A Scanning Tunneling Microscope at the Milli-Kelvin, High Magnetic Field Frontier
Brian B. Zhou
A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy
Recommended for Acceptance by the Department of Physics Advisor: Ali Yazdani
September 2014
c Copyright by Brian B. Zhou, 2014.
All rights reserved.
Abstract The ability to access lower temperatures and higher magnetic fields has precipitated breakthroughs in our understanding of physical matter, revealing novel effects such as superconductivity, the integer and fractional quantum Hall effects, and single spin magnetism. Extending the scanning tunneling microscope (STM) to the extremity of the B-T phase space provides unique insight on these phenomena both at the atomic level and with spectroscopic power. In this thesis, I describe the design and operation of a full-featured, dilution refrigerator-based STM capable of sample preparation in ultra-high vacuum (UHV) and spectroscopic mapping with an electronic temperature of 240 mK in fields up to 14 T. I detail technical solutions to overcome the stringent requirements on vibration isolation, electronic noise, and mechanical design necessary to successfully integrate the triad of the STM, UHV, and dilution refrigeration. Measurements of the heavy fermion superconductor CeCoIn5 (Tc = 2.3 K) directly leverage the resulting combination of ultra-low temperature and atomic resolution to identify its Cooper pairing to be of dx2 −y2 symmetry. Spectroscopic and quasiparticle interference measurements isolate a Kondo-hybridized, heavy effective mass band near the Fermi level, from which nodal superconductivity emerges in CeCoIn5 in coexistence with an independent pseudogap. Secondly, the versatility of this instrument is demonstrated through measurements of the three-dimensional Dirac semimetal Cd3 As2 up to the maximum magnetic field. Through high resolution Landau level spectroscopy, the dispersion of the conduction band is shown to be Dirac-like over an unexpectedly extended regime, and its two-fold degeneracy to be lifted in field through a combination of orbital and Zeeman effects. Indeed, these two experiments on CeCoIn5 and Cd3 As2 glimpse the new era of nano-scale materials research, spanning superconductivity, topological properties, and single spin phenomena, made possible by the advance of STM instrumentation to the milli-Kelvin, high magnetic field frontier. iii
Acknowledgements To begin, I would like to thank my advisor, Prof. Ali Yazdani, for providing me the opportunity to work on the projects presented in this thesis. Ali’s steady leadership and motto of focusing on “how far we have come” rather than “how far we need to go” was one driving force in carrying this long journey through the toughest of times. It has been amazing to witness and learn from Ali’s special ability to communicate delicate physical concepts in scientific writing, presentation, and discussion. No beginning graduate student could undertake alone the job of constructing a dilution fridge STM (or more colloquially DRSTM), and tremendous credit for this thesis should go to post-doc Shashank Misra, who mentored me in the experimental aspects over the 5 plus years we spent together debugging the instrument. Shashank’s influence will be long engrained in the users of DRSTM, from his many clever designs, meticulous protocols, to even the password on the measurement computer. I have enjoyed fruitful and instructive collaborations with the groups of Dr. Eric Bauer, Prof. Bob Cava, and Prof. Ashvin Vishwanath. Staff members in the Princeton physics department, including Steve Lowe, Bill Dix, Darryl Johnson, Ted Lewis, James Kukon, Claude Champagne and many others, have facillated the research in this thesis with great kindness. I would also like to thank Prof. Waseem Bakr for his reading of this thesis, and Prof. Nai Phuan Ong and Prof. Mariangela Lisanti for serving on my oral committee. In addition, my long tenure in Ali’s lab has allowed me to bridge two eras of lab members, whose camaraderie has made the time here fly by. When I first arrived, I looked up to the elder generation of Kenjiro, Aakash, Anthony, Lukas, Pedram, and Colin, as they set fine examples in the lab. Concurrent to me were the proverbial “good physicists” of Eduardo, Jungpil, Pegor, Haim, Andras, Ilya, and Stevan. Nowadays, I applaud the new crew of Sangjun, Mallika, Yonglong, and Ben for being eager and able to continue the tradition. Sangjun’s diligent effort and push to explore iv
multiple interpretations of the Cd3 As2 data was a huge factor in our overall efficiency and led to greater understanding of the basic model. Last and foremost, I am deeply grateful for the care, lessons, and encouragement throughout my life from my parents Lingling and Genwen, to whom I dedicate this thesis.
v
Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
1 Introduction
1
1.1
Scanning Tunneling Microscopy at the Limit . . . . . . . . . . . . . .
2
1.2
Heavy Fermions - The Kondo Lattice . . . . . . . . . . . . . . . . . .
6
1.3
Heavy Fermions - Unconventional Superconductivity . . . . . . . . .
11
1.4
Three-Dimensional Dirac/Weyl Semimetals . . . . . . . . . . . . . . .
17
1.5
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2 The Basics of Scanning Tunneling Microscopy
25
2.1
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2
Quasiparticle Interference Imaging . . . . . . . . . . . . . . . . . . . .
29
3 The Dilution Refrigerator Scanning Tunneling Microscope
35
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.2
Ultra-High Vacuum Assembly . . . . . . . . . . . . . . . . . . . . . .
37
3.3
Vibration Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.4
The Dilution Refrigerator and Microscope Head . . . . . . . . . . . .
43
3.5
STM Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
vi
3.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
4 Visualizing d-Wave Heavy Fermion Superconductivity in CeCoIn5
54
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.2
Superconductivity on the Two Surfaces of CeCoIn5 . . . . . . . . . .
56
4.3
Quasiparticle Interference in Normal and Superconducting States of CeCoIn5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4.4
Response of Nodal Superconductivity to Potential Scattering . . . . .
62
4.5
Vortex Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.6
Impurity Bound State: Fingerprint of dx2 −y2 Pairing . . . . . . . . . .
66
4.7
Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
5 The Three Dimensional Dirac Semimetal Cd3 As2
69
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
5.2
Topographic and Spectroscopic Characterization at Zero Field . . . .
71
5.3
Landau Level Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . .
74
5.4
Spatial Homogeneity of Landau Levels . . . . . . . . . . . . . . . . .
78
5.5
Quasiparticle Interference . . . . . . . . . . . . . . . . . . . . . . . .
78
5.6
Landau Level Simulation . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.7
Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
6 Conclusion
84
A Further Experimental Aspects of DRSTM
90
A.1 X and Z Capacitance Position Sensors . . . . . . . . . . . . . . . . .
90
A.2 Life on DR: Including Approaching an Sample . . . . . . . . . . . . .
93
A.3 ‘Joule-Thomson’ 2K Mode Operation . . . . . . . . . . . . . . . . . .
95
A.4 Dewar Exhaust Management . . . . . . . . . . . . . . . . . . . . . . .
97
A.5 Electrical Ground Loop Management . . . . . . . . . . . . . . . . . .
99
vii
B Multipass Spectroscopy: An Alternative to Conventional Conductance Spectroscopy B.1 Traditional Conductance Maps
102 . . . . . . . . . . . . . . . . . . . . . 102
B.2 The Multpass Technique . . . . . . . . . . . . . . . . . . . . . . . . . 104 B.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 C Comparison of QPI in CeCoIn5 to Other Band Structure Probes and Phenomenological Modeling
108
C.1 Reference to Other Experimental Mappings of the Band Structure . . 108 C.2 Phenomenological Modeling of Normal State Band Structure . . . . . 110 C.3 Superconductivity Gapping the Phenomenological Band Structure . . 112 D Details of Cd3 As2 Landau Level Simulation
116
D.1 Modified Four-Band Kane Hamiltonian . . . . . . . . . . . . . . . . . 116 D.2 Schematic Demonstration of the Weyl Fermion and the Low Field Regime124 E Charge Ordering in Underdoped Bi2 Sr2 CaCu2 O8+δ in a Magnetic Field
126
E.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 E.2 Energy and Spatially-Resolved Density of States in a Magnetic Field
129
E.3 Where are the Vortices? . . . . . . . . . . . . . . . . . . . . . . . . . 133 Bibliography
136
viii
List of Tables C.1 Estimated minimal Q vectors from dHvA measurements of CeCoIn5 . . 110 D.1 Parameters for the modified four-band model for Cd3 As2 . . . . . . . . 123
ix
List of Figures 1.1
Goal of Thesis Project . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2
Cyrogenic Scanned Probe Instruments . . . . . . . . . . . . . . . . .
5
1.3
The Heavy Fermion Phase Diagram . . . . . . . . . . . . . . . . . . .
8
1.4
Tunneling into the Two Surfaces of CeCoIn5 . . . . . . . . . . . . . .
11
1.5
Pairing Symmetry of Superconductivity . . . . . . . . . . . . . . . . .
12
1.6
Antiferromagnetic Correlations on a Square Lattice . . . . . . . . . .
16
1.7
Three Dimensional Dirac Semimetal in a Magnetic Field . . . . . . .
19
2.1
The Tunneling Current from Density of States . . . . . . . . . . . . .
27
2.2
Modes of STM Operation . . . . . . . . . . . . . . . . . . . . . . . .
29
2.3
Demonstration of QPI for the Cu(111) Surface State . . . . . . . . .
31
3.1
General Assemby of DRSTM . . . . . . . . . . . . . . . . . . . . . . .
38
3.2
The Ultra-Quiet Laboratory . . . . . . . . . . . . . . . . . . . . . . .
41
3.3
Vibration Performance . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.4
Microscope Design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.5
Aluminum Superconducting Gap . . . . . . . . . . . . . . . . . . . .
48
3.6
Spectroscopic Mapping Performance . . . . . . . . . . . . . . . . . . .
49
3.7
Tunneling Current Noise Characteristic . . . . . . . . . . . . . . . . .
51
4.1
Hybridization, Pseudogap, and Superconductivity . . . . . . . . . . .
56
4.2
First Order Phase Transition at Hc2 . . . . . . . . . . . . . . . . . . .
58
x
4.3
Quasiparticle Interference of Heavy Superconducting Electrons . . . .
60
4.4
Enhancement of Q3 Vector Along (π, π) Nodal Direction . . . . . . .
62
4.5
Evolution of In-Gap Quasiparticle States Approaching a Step-Edge. .
63
4.6
Superconducting Gap Approaching an Step-Edge in a s-Wave Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.7
Vortex Lattice and Anisotropy of Bound State in CeCoIn5 . . . . . .
65
4.8
Impurity-Bound Quasiparticle Excitations in a dx2 −y2 Superconductor.
67
4.9
Normalized Spectrum at Center of Impurity. . . . . . . . . . . . . . .
68
5.1
STM Characterization of Cd3 As2 . . . . . . . . . . . . . . . . . . . .
73
5.2
Conductance Fluctuation of Defects in As Plane . . . . . . . . . . . .
74
5.3
Landau Level Spectroscopy of Cd3 As2 . . . . . . . . . . . . . . . . . .
76
5.4
Quasiparticle Interference in Cd3 As2 . . . . . . . . . . . . . . . . . .
79
5.5
Landau Level Simulation for Cd3 As2 . . . . . . . . . . . . . . . . . .
81
6.1
The Phase Diagram of CeCoIn5 with Doping . . . . . . . . . . . . . .
86
6.2
Surface Atom Ordering in (112) Plane of Cleaved Cd3 As2 . . . . . . .
88
A.1 X and Z Capacitance Sensors . . . . . . . . . . . . . . . . . . . . . .
91
A.2 Model Tip Sample Capacitance . . . . . . . . . . . . . . . . . . . . .
94
A.3 Joule-Thomson Mode Characteristics . . . . . . . . . . . . . . . . . .
97
A.4 Dewar Exhaust Flow Chart . . . . . . . . . . . . . . . . . . . . . . .
98
A.5 Electrical Wiring Chart . . . . . . . . . . . . . . . . . . . . . . . . . .
99
B.1 Traditional Measurement Cycle for Conductance Maps . . . . . . . . 103 B.2 Schematic of Multipass Technique for Acquiring Conductance Maps . 104 C.1 Comparison of QPI to Fermi Surface from ARPES and Theoretical Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 C.2 Tight Binding Bandstructure To Experimental Dispersion xi
. . . . . . 111
C.3 Superconducting QPI in comparison to simulation of dx2 −y2 and dxy gap symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 D.1 Model Band Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 124 D.2 Demonstration of Weyl Fermion . . . . . . . . . . . . . . . . . . . . . 125 E.1 Bi2212 (UD 58 K) in a magnetic field . . . . . . . . . . . . . . . . . . 128 E.2 Charge Ordering with a Magnetic Field . . . . . . . . . . . . . . . . . 130 E.3 Fourier Transform of In-Field Spatial Patterns . . . . . . . . . . . . . 132 E.4 Vortex regions in underdoped Bi2212 . . . . . . . . . . . . . . . . . . 135
xii
Chapter 1 Introduction The full splendor of the phases and phenomena of matter cannot be appreciated by our everyday experience, one that traverses only a small confined area in the phase space of temperature, magnetic field, and pressure - much like the full richness of a symphony cannot be appreciated from listening to only a restricted section of an orchestra. Throughout the history of science, the advancement of measurement techniques to lower temperature and higher magnetic fields has expanded our perception to the far-ranging nuances of this symphony with mother nature as its maestro. From the liquefaction of helium by Kamerlingh Onnes in 1908 came his discovery of superconductivity in elemental mercury (1911) [1] and thereafter the discovery of superfluidity in liquid helium-4 (1937) by Pyotr Kapitsa and John F. Allen [2]. Later experiments on two dimensional electron gases in strong magnetic fields and at low temperature demonstrated that resistance under these extreme conditions is quantized in integer and fractional divisions of the fundamental value RH = h/e2 [3]. Indeed, these observations at extreme conditions, which contrast with our ordinary experience, profoundly advanced our understanding of the details and consequences of quantum physics, from degenerate quantum wavefunctions, the interplay of localization and dimensionality, to topological invariance. Guided by this principle, 1
this thesis details an effort to extend the scanning tunneling microscope (STM) to the extreme limit of milli-Kelvin temperatures and very high magnetic fields. The success of this effort is demonstrated through two experiments that utilize the new instrument’s access to an expanded phase space to probe unconventional superconductivity in the heavy fermion material CeCoIn5 and the exotic band structure of the three-dimensional Dirac semimetal Cd3 As2 . In this introduction, we overview the capabilities of the STM measurement technique and the current state of the technology in this field. Finally, we introduce the core concepts of heavy fermion materials and Dirac/Weyl semimetals and preface the experimental findings presented in this thesis.
1.1
Scanning Tunneling Microscopy at the Limit
If a picture is worth a thousand words, then the scanning tunneling microscope has spoken volumes about the quantum world, illustrating seemingly invisible concepts in plain view, from the existence of atoms, the wave nature of electrons, to the pairing symmetry of superconductivity. Indeed, Binnig and Rohrer’s invention of the STM at IBM Zurich immediately clarified a two decade long mystery about the precise nature of the 7x7 reconstruction on the Si (111) surface [4]. The ability to trace the quantum mechanical tunneling current as a sharp metal tip is scanned within 10 ˚ A of a clean material surface heralded an new age of atomic resolution. Moreover, by recording this current while ramping the chemical potential of the sample with respect to the tip, the electronic density of states of the sample, both above and below the Fermi energy, can be probed with precision, reliability, and in a magnetic field. While the precise details of STM operation will be discussed in Chapter 2, it suffices now for the reader to appreciate the power of resolving electronic structure at the local level over only having information about its bulk behavior. For example, the ability to correlate 2
electronic inhomogeneity to the local atomic structure, such as strain or chemical substitution, enables the direct connection between structure and function, providing direction for changing material properties [5]. Second, phase transitions, such as superconductivity, anti-ferromagnetism or charge order, oftentimes nucleate locally before globally, a process which reflects the correlations that drive the phase transition [6]. Finally, the response of a transitionally-invariant state to local perturbations, such as superconductivity to a pair-breaking impurity, reveals the fabric of the original state, much like how ripples from a stone dropped on a lake reveals information about the properties of water [7]. Extending these powerful STM measurements to the lowest temperatures, such as that provided by a dilution refrigerator, and to the highest magnetic fields, provided by a superconducting cryo-magnet, represents the objective of this thesis work. While a long standing goal in the STM community, the marriage of such cryogenic, magnetic, and STM hardware, however, puts several essential technical requirements in conflict with each other: 1) while the STM requires sub-picometer vibrational stability, running a dilution refrigerator requires mechanical pumps which couple to the STM instrument via rigid pumping lines and via the internal flow of cryogen, 2) the most powerful STM experiments require atomically-clean surfaces, which must be prepared in ultra-high vacuum (UHV) either through cleaving or cleaning procedures and be subsequently transferred to the cold STM head without interruption of UHV, and 3) high magnetic fields can induce both static and dynamic picometer deflections due to any residual magnetism in the STM head. Thus, requirement 2) demands that the dilution fridge be carefully incorporated into the UHV environment, necessitating innovative mechanical design to facilitate the sample transfer, while together 2) and 3) necessitate careful choice of UHV-compatible, non-magnetic materials in the construction of the entire fridge and STM head. However, the first requirement of picometer vibrational stability while maintaining a closed cycle of cryogen in the 3
B
T
Figure 1.1: The goal of the project described in this thesis is to extend the powers of the STM, from visualizing single atoms to the wave nature of electrons, to the lowest temperature and the highest magnetic fields. The feat to accomplish is to adapt the generically “table-top” STM (photo courtesy Kenjiro Gomes) and its UHV environment to be compatible with a dilution refrigerator and superconducting magnet. presence of an internal liquid reservoir (the “1K pot”) turns out to be the foremost challenge [8]. This thesis details solutions, some complete, some partial, both in design and implementation, that have overcome the challenges faced to demonstrate high performance scanning tunneling microscopy at dilution fridge temperatures and high magnetic fields. The instrument was constructed in the lab of Prof. Ali Yazdani at Princeton University over the period of five years starting from the summer 2008 to the summer of 2013 [10]. The figure of merit can be taken as the ratio of the mag4
UHV (This Thesis) True B/Telectron
Princeton
Figure 1.2: The figure of merit µB B/kB T in logarithmic scale for notable cryogenic scanning probe instruments as a function of year of demonstration, where T is the minimum lattice temperature and B is the maximum magnetic field of the instrument’s operation. Red circles denote instruments compatible with ultra-high vacuum, while blue diamonds denote cryo-vacuum systems. The Princeton UHV dilfridge-based STM system described in this thesis is shown as the orange circle, while the the orange circle with black outline denotes for this system the more operational metric µB B/kB Telectron . Figure adapted from [9]. netic to thermal energy scales µB B/kB T , which can be regarded as a measure of the spectral resolution for magnetic properties or as a measure of the accessible area in B ∗ 1/T phase space. As shown in Fig. 1.2, only recently have a handful of high field, cryogenic STMs have come into existence to reach the highest µB B/kB T metrics. We highlight the dilution fridge-based instruments represented by the data points labeled “NIST” at the National Institute of Standards and Technology in Maryland [9, 11] and labeled “Univ. Auton. Madrid” at the Universidad Autnoma de Madrid, Spain [12], as they have demonstrated scientific measurements in addition to instrument characterization. However, the instrument at Madrid is restricted to studying only cleaved samples, kept clean by cryo-vacuum, rather than the full set of metallic samples available by UHV cleaning procedures. The UHV instrument at Princeton 5
constructed during the course of this thesis has demonstrated scientific measurements at a temperature of 20 mK and a magnetic field of 14 T, yielding a nominal µB B/kB T ∼ 470, where T denotes the lattice temperature. While using the lattice temperature is the fair comparison to the other data points, we show for reference on the same graph the more practical metric µB B/kB Telectron ∼ 14 T/240 mK ∼ 40 for the Princeton system. The distinction between electronic and lattice temperature will be further discussed in Chapter 3. Regardless of the metric, the ultimate value of our instrument lies in the physical insights it can reveal, and we now transition to describing two experiments that have benefited from its enhanced capabilities.
1.2
Heavy Fermions - The Kondo Lattice
A central focus of condensed matter physics is the study of the collective behavior of electrons in a solid. The interactions of electrons with other electrons and protons in a crystal lattice, both through charge and spin degrees of freedom, can modify their behavior from the non-interacting, free-space description ε(k) = ~2 k 2 /2m0 , where m0 is the bare electron mass. It is precisely these interactions that drive the rich variety of emergent solid state behavior, such as the Mott insulator or ‘heavy fermion’, and lead to technologically useful physical properties, such as colossal magnetoresistance and high temperature superconductivity. In the presence of interactions, Landau Fermi liquid theory can describe the new correlated state in terms of quasiparticles, or new emergent excitations that represent a collective motion of the many-body system. These excitations can be treated in certain limits as independent quasi particles which have an renormalized mass m∗ and a renormalized magnetic moment µ∗ . The low energy quasiparticles in heavy fermion compounds [13, 14, 15] possess renormalized masses m∗ up to 1000 times the free electron mass m0 as a result of the Kondo effect, to be described below. Experimentally, since m∗ is proportional to the 6
density of states at the Fermi level N (0), the physical observables of heavy fermion formation are the corresponding thousand-fold enhancements of the electronic specific heat (γ) and the Pauli paramagnetic susceptibility (χ), both proportional to N (0), beneath a characteristic temperature scale T ∗ . Compounds that display heavy fermion behavior are intermetallics containing partially filled 4f and 5f elements, predominantly the four elements Ce, Yb, U, and Np. Indeed, as exemplified by Ce and Yb, which occupy at the extremes of the lanthanide block, possessing respectively have a single 4f electron and a completely filled 4f shell in the atomic limit, heavy fermion formation is driven by the instability between localized magnetic moments and electronic conduction. For these critical f -elements, the balance between the two can be tipped if the occupation of a single electron switches between a localized, magnetic f shell or a delocalized, bonding spd shell depending on its chemical environment of the intermetallic compound. In actuality, this critical electron will often occupy a hybridized orbital with both f - and spd- character. In this sense, heavy fermions lie at the brink of magnetism, where the enhanced and tunable interplay between electronic and magnetic degrees of freedom produce remarkable physics such as quantum criticality and unconventional superconductivity. Formally, we can first consider the propensity of local f -electrons to participate in electronic conduction through the Anderson impurity model for a single magnetic ion dissolved in a conduction sea:
H=
X k,σ
|
k nk,σ +
X k,σ
V (k) c†k,σ fσ + fσ† ck,σ + Ef nf + U nf ↑ nf ↓ , | {z } f Coulomb repulsion {z }
(1.1)
f-c hybridization
where the first set of terms describes the hybridization of the magnetic f -level to the conduction sea via the interaction V (k) and the second set of terms captures the Coulomb physics of local moment formation for the single f -level [13]. Analysis of this Hamiltonian by Jun Kondo and others led to the understanding of the celebrated 7
Kondo effect, where it was shown that a singlet wavefunction, with anti-ferromagnetic coupling of the conduction sea to the local f moment, results in an energy gain, or equivalently temperature scale TK , 1 TK = D exp − , N (0)|J|
(1.2)
where J ∼ V (k)2 reflects the exchange coupling between the conduction sea and the localized moment and D reflects the conduction bandwidth. Thus below TK , the conduction electrons screen the local f moment via hybridization, resulting in a low energy resonance in their density of states effectively arising from entanglement to the spin degeneracy of the f -ion.
a)
b)
Heavy Fermion
T
CeCoIn5
TRKKY TKondo “Non-Fermi Liquid” AF Order
0
Fermi Liquid
δ
δc Superconductivity
Figure 1.3: a) Doniach phase diagram for a generic heavy fermion, showing the competition between the Kondo and RKKY interactions as a function of a tuning parameter δ, which sets the strength of hybridization. b) In the CeCoIn5 phase diagram, the rich variety of phases can be tuned through either pressure, magnetic field, or chemical doping with Cd atoms. Figure adapted from [16]. To describe heavy fermion systems, equation 1.1 must be extended from the dilute single ion limit to the dense limit of a lattice of f -level ions via the so called periodic Anderson Hamiltonian. In this case, the Kondo temperature TK is renormalized, but maintains a similar function. Instead of a single Kondo resonance, a heavy 8
fermion band of resonances, incorporating the spin entropy of each f -lattice site, forms at low energies, and produces the huge density of states at the Fermi level that is the hallmark of heavy fermion properties. In general, the lattice Kondo effect competes with the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, which is also mediated by the conduction electrons, but tends to order magnetic ions in a lattice antiferromagnetically (AF), rather than compensating the magnetic ions as in the Kondo interaction. The energy gain of the RKKY interaction scales as a power law in exchange coupling J via
TRKKY = N (0)J 2 .
(1.3)
Hence as shown in Fig. 1.3 a), in the limit of small hybridization J, the AF ordered, local moment state is favored since the RKKY energy gain dominates the Kondo energy gain, while at high hybridization, the Kondo-screened Fermi liquid is favored [13]. This competition underlies the heavy fermion phase diagram, where a quantum critical point (QCP) separates the AF and Fermi-liquid states at zero temperature and can be explored through tuning the strength of hybridization, via external pressure, applied magnetic field, and chemical substitution. Moreover, a dome of superconductivity oftentimes occurs near the QCP where magnetic correlations are the strongest, just as the Kondo and RKKY interactions balance. In the prototypical heavy fermion CeCoIn5 studied in this thesis, all three variables of pressure, field, and doping can change the ground state phase, sometimes even reversibly - for example, Cd-doping transitions the system towards AF, suppressing superconductivity, but the application of pressure can revert the Cd-doped material back to superconductivity [17]. STM spectroscopic techniques were first extended to the physics of heavy fermion compounds for URu2 Si2 in Refs. [18] and [19]. In URu2 Si2 , however, the unexplained hidden order phase with temperature scale THO = 17.5 K [20] interrupts the Kondo 9
physics that onset eariler around T ∗ ∼ 70 K and thus complicates interpretation of the measurement. In 2012, through variable temperature experiments, Aynajian and da Silva Neto [21] used quasiparticle interference (QPI) to definitively visualize the formation of heavy fermions in CeCoIn5 , a model Kondo lattice system. These authors showed that the weakly dispersing light band structure is transformed below T ∗ into a strongly dispersing heavy fermion band. Furthermore, in Ref. [21], it was found that upon cleaving CeCoIn5 , different surface terminations (layers) are exposed, and the local tunneling spectra measured by STM depend on which surface termination is measured. This effect was explained in terms of the interference of the two tunneling paths, one into the lighter part of the composite heavy quasiparticle (the spd-like band) and the other into the heavier, more localized part (the f -resonance). One of the two atomically ordered surfaces of CeCoIn5 , “surface A” was identified as the CeIn surface, and the tunneling spectra show a pronounced gap near the Fermi energy, reflective of the hybridization gap in the spd-like band. In contrast, on “surface B”, identified as the Co surface, the spectra show a peak near the Fermi energy, reflective of the accumulation of the heavy fermion density of states caused by the participation of the f -electrons in conduction. The work in this thesis [7] builds previous work by performing higher resolution measurements at the lowest temperature (250 mK) on surface B of CeCoIn5 .1 Here, utilizing the enhanced tunneling sensitivity to the heavy quasiparticles, these experiments better resolve the energy-momentum structure of the heavy fermion band to enable quantitative comparisons to band structure calculations. However, even more excitingly, the lower temperature allows access to the dome of superconductivity, whose proximity to magnetic fluctuations is thought to mediate unconventional Cooper pairing symmetries. This interplay of magnetism and superconductivity ties the heavy fermion compounds to the high temperature cuprate superconductors, and 1
Ref. [21] performed QPI on surface A.
10
a)
b)
c) Ce-In In Co
Figure 1.4: The pioneering experiments of Ref. [21] showed CeCoIn5 to cleave along [001], exposing multiple different surfaces, most notably ‘surface’ A (the Ce-In layer) and ‘surface B’ (the Co layer). Because of different tunneling matrix elements on the two different surfaces, the measured density of states reflects the hybridization of the band structure in different manners. On surface A, the density of states become gapped beneath TK , as the spectral weight of the light electrons (green band in a)) is lost as they hybridize and become heavy near the Fermi energy. On surface B, the counterpart of the hybridization process is revealed: the onset of heavy quasiparticles (red band in a)) results in a peak in the density of states. its understanding may provide the essential link between magnetic correlations and higher Tc ’s.
1.3
Heavy Fermions - Unconventional Superconductivity
Superconductors, broadly speaking, can be divided into two classes by how its electrons bind together to form the Cooper pairs that sustain dissipation-less current 11
flow. The first class of conventional superconductors contains all of the metallic superconductors, such as mercury, lead, or niobium. Here the attractive potential, or pairing symmetry (represented by a circle in Fig. 1.5), of the Cooper pair is equally strong for all electrons, and the transition temperature Tc at which superconductivity emerges is limited to at most 30 K, only one tenth of room temperature. The second class of unconventional, or extraordinary, superconductors behave much differently and strikingly can possess transition temperatures in excess of 150 K, half of room temperature. The pairing potential (represented by a four-leaf clover in Fig. 1.5) now depends sensitively on the direction of the electron’s momentum, with certain directions (called nodes) having zero pairing strength.
s-wave ky
d-wave
∆
kx
ky
kx
Figure 1.5: Left: Representation of the pairing symmetry of a conventional superconductor by a circle. The pairing amplitude is isotropic in direction. Right: Representation of the pairing symmetry for an unconventional dx2 −y2 superconductor by a clover shape. The pairing amplitude is zero for certain directions called the nodes, depicted by the dashed lines. The route to higher temperature superconductors relies on understanding the ingredients that support strong Cooper pairing. The pairing symmetry (or order parameter) of a superconductor is the key experimental observable that manifests the underlying mechanism which binds electrons together. The conventional superconductors are well explained by a phonon-mediated interaction under the BardeenCooper-Schrieffer (BCS) theory that assumes the simplest k-independent, s-wave or12
der parameter [22, 23]. The conceptual breakthrough of BCS theory is that the range of the Coulomb repulsion between two spatially separated electrons can be reduced greatly by screening in metal, to the point that the electron’s joint interaction with a longer-lived lattice deformation of the ionic cores can create a stronger effective attraction. Thereafter, the Fermi sea is unstable to any arbitrarily weak attraction, choosing instead to form bound electron pairs of the |k, ↑i and |−k, ↓i states, called Cooper pairs, which condense into a macroscopic wavefunction at sufficiently low temperature and give rise to the phenomenal superconducting properties of zero resistance and the Meissner effect. Indeed, the electron-phonon mechanism proposed by BCS theory is well verified experimentally through the isotope effect, which showed that the Tc ’s of elemental isotopes of Hg, for example, scaled inversely with the square root of their nuclear mass, a parameter which only tunes the lattice properties [24]. In addition to the phonon-mediated electron attraction, theorists believe that a magnetically mediated interaction, based on fluctuations in the electron’s spin degree of freedom, can also provide the attractive force necessary for Cooper pairing and can in contrast produce more exotic, anisotropic, finite angular momentum p- or dwave order parameters. These ‘magnetic’ theories trace their origin to the superfluid phase of 3 He, which can also be considered as the result of Cooper-type pairing of the helium fermions into “extended molecules”. While the attractive interaction in this case is of van der Waals origin, rather than spin-mediated, the critical concept is that unconventional, finite angular momentum p-wave pairing (l = 1) can minimize the direct on-site Coulomb repulsion between two helium nuclei to allow the attractive interaction to be effective [25]. Thereafter, these ideas were extended to explain the anomalous superconducting signatures in the organic and heavy fermion compounds in the early 1980s. In the heavy fermion compound UPt3 , d-wave pairing was first proposed as the outcome of a spin density wave instability for the Hubbard model on a cubic lattice [26]. However, the prominence of such ‘magnetic’ theories exploded 13
after the shocking discovery of high temperature superconductivity in 1986 in the doped copper oxides, whose transition temperature ∼100 K eclipsed the previously known superconductors by an order of magnitude. The key distinction of the cuprate superconductors is that the zero doping parent state is an anti-ferromagnetically ordered Mott insulator, the model system for the Hubbard Hamiltonian with strong on-site repulsion U . Indeed, both measurements sensitive to the amplitude of the order parameter (angle-resolved photoemission (ARPES), angle-resolved transport studies) and more powerfully, to the phase (i.e., sign) of the order parameter (grain boundary Josephson junctions) firmly establish the order parameter symmetry of the cuprate superconductors to be dx2 −y2 , thus lending strong support to a magneticallymediated mechanism of superconductivity [27]. It is useful to motivate at least a partial understanding of how unconventional d-wave pairing can be plausible, particularly in the case of spin fluctuations on a square lattice; however, a rigorous calculation of the origin of this magnetic attractive interaction is nontrivial and may depend on the details of the particular band and crystal structure. In general, we can still consider solutions to the BCS gap equation [28]:
∆k = −
∆k0 1 X Vkk0 p 2 N k0 2 ξk0 + ∆2k0
(1.4)
where Vkk0 is the pairing interaction, ξk is the electronic dispersion referenced with respect to the Fermi energy, and ∆k is the gap function. For an k-independent, attractive Vkk0 = −V0 , an isotropic, BCS-type s-wave ∆k = ∆0 solution to equation 1.4 can be verified to exist. On the other hand, investigations of the Hubbard model on a square lattice showed that in the limit of large on-site repulsion, the effective interaction can be treated as a spin-spin interaction between nearest neighbors with an antiferromagnetic J, stemming from virtual hopping processes, and leads to a
14
ground-state of nearest neighbor AF-ordered spins as shown in Fig. 1.6. When this ground state is doped with holes, it is plausible that an attractive2 nearest neighbor interaction for the holes can arise, having the generic form
Vkk0 = 2 V1 cos (kx − kx0 )a + cos (ky − ky0 )a
(1.5)
with V1 < 0 and a as the lattice constant for a square lattice. From this one immediately realizes the BCS gap equation in 1.4 can be satisfied if ∆k maintains its sign (changes adiabatically) when k − k0 → (0, 0) (and Vkk0 < 0), but flips its sign when k − k0 → (π/a, π/a) (and Vkk0 > 0). These criteria and the four-fold symmetry of the lattice then motivate the dx2 −y2 gap function diagrammed in Fig. 1.5
∆k = ∆0 (cos(kx a) − cos(ky a))
(1.6)
which satisfies the the condition of continuity for small displacements along the Fermi surface and the condition for sign change for parts of the Fermi surface connected by the vector Q = (π/a, π/a). Moreover, the cuprate Fermi surfaces contain long parallel sections nested by the vector Q, which maximizes the the advantage of such a dx2 −y2 gap function. In summary, a nearest neighbor attractive interaction, plausibly mediated by AF spin fluctuations with a ordering vector ±Q, and a favorable Fermi surface shape can provide the ripe background for unconventional dx2 −y2 pairing. The heavy fermion CeCoIn5 has been suggested to share the same pairing symmetry and thus possibly the same underlying mechanism of superconductivity due to its many other similarities to the cuprate superconductors, such as its tetragonal lattice structure with quasi-2D square lattice planes, its proximity to an AF critical point, and its non-Fermi liquid normal state.3 Unfortunately, its superconducting 2
We have not proven that it is attractive, but will examine the consequences of this assumption. It is also important to keep in mind the disparities between the heavy fermion and cuprate compounds. The cuprates are more itinerant, d-electron compounds with strictly two-dimensional 3
15
Q +
U
+
-
t J = 4t2/U
Figure 1.6: The Hubbard model, characterized by an on site repulsion U and a hopping energy t, on a square lattice leads to an antiferromagnetic arrangement of spins at half filling. When holes are doped into the system, remnant antiferrogmagnetic correlations could lead to an attractive interaction between the now mobile electrons. When this interaction combines with a Fermi surface with nested (parallel) sections separated by the ordering vector Q = (π/a, π/a) , dx2 −y2 superconductivity is believed to be favored. phase has largely dodged the same intense experimental spotlight, as its low transition temperature (Tc = 2.3 K) precludes access by conventional experimental tools such as ARPES and, until recently, STM. While angle-resolved thermal transport [29] and neutron scattering experiments [30] have shown data consistent with dx2 −y2 gap symmetry, no smoking gun identification has yet been reported. In CeCoIn5 , more favorable to experiment, unconventional superconductivity occurs in the undoped, and thus ultra-clean, samples and can be extinguished by an experimentally-accessible magnetic field (5 T perpendicular to the c-axis). This thesis details a comprehensive set of STM experiments performed by Zhou and Misra [7], spanning spectroscopy and quasiparticle interference, on the superconducting state of CeCoIn5 at the ultra-low electron temperature of 250 mK. To isolate the salient effects of superconductivity, we in fact conducted each experiment first conduction and a Mott insulating parent state. In some sense, the reduced dimensionality and strong Mott repulsion offsets the greater itinerancy, in comparison to the f -electron, more three dimensional heavy fermion superconductors. Moreover, the bandwidths t ∼ V in the cuprate superconductors are much larger than in the heavy fermion compounds (t ∼ 10 mV), but the ratio Tc /t for the two families are comparable.
16
at zero magnetic field, where superconductivity is strongest, and then repeated each experiment again at 5.7 T, where superconductivity is extinguished. Unexpectedly, QPI band mapping demonstrates particle-hole asymmetric patterns in the superconducting state, reflective of either the more rapid heavy fermion band dispersion or enhanced impurity effects relative to that for the cuprate superconductors. However, likewise to the cuprates, a spectroscopic pseudogap appears prior to the onset of superconductivity and persists above the critical magnetic field. Finally, through visualizing the spatial symmetry of quasiparticle states bound to atomic defects, the structure of Cooper pairing in CeCoIn5 is pinpointed to be dx2 −y2 , in parallel with that of the high-temperature cuprate superconductors. Long term, the ability to study CeCoIn5 broadens the experimental tool-kit for tackling the questions of unconventional superconductivity: what role of magnetism, other competing phases, and electron-electron interaction in the normal state play in making these special superconductors the most robust of the bunch.
1.4
Three-Dimensional Dirac/Weyl Semimetals
The notion of topologically protected physical effects that manifest without fine tuning and survive in the presence of disorder is a welcome concept to the experimental physicist, trained to pay attention to the finest details. Tracing its origins to the integer quantum Hall effect [3], where exquisitely precise resistance quantization (∼ one part in a billion) literally thrives because of disorder, this idea that certain effects stem from the intrinsic topology of the system and are thus robust to perturbations that do not violate the original symmetries, has flourished in recent years with the discovery of topological insulators, whose band structure guarantees the existence of surface states at its interface [31]. Recently, another example of a topologically protected phase, this time in a semimetal rather than insulator, has been proposed: 17
the Weyl semimetal, whose topological band touchings give rise to similarly protected surface states, known as ‘Fermi’ arcs, and to the condensed matter analogue of the chiral anomaly, which resolved the paradox of the decay of the π 0 meson in high energy physics [32]. Weyl semimetals contain discrete points in its Brillouin zone where two nondegenerate bands touch, in contrast to Dirac semimetals where inversion and time reversal symmetries guarantee the double-degeneracy of each of the touching bands [33, 34]. Hence, to realize a Weyl semimetal, either time reversal or inversion symmetry must be broken, for example, by a magnetic field or by a layered heterostructure scheme, respectively. The reduction of dimensionality from the 4x4 space used to describe the physics around a Dirac point to the 2x2 space around a Weyl point presents important consequences for the robustness and topology of Weyl points. Generically, the low energies dispersions around a Weyl point is linear in three dimensions and can be described by the most general linearly-coupled Hamiltonian
H(k0 + q) = v0 · q 1 +
3 X
vi · q σi
(1.7)
i=1
where q is the displacement from the Weyl node at k0 , the identity matrix 1 and the Paul matrices σi span the space of 2x2 Hermitian matrices, and each of the four three-dimensional velocities vi linearly couple to the momenta q = (qx , qy , qz ). The simplest reduction of this Hamiltonian is
H(k0 + q) = ±vF q · σ = ±vF (qx σx + qy σy + qz σz )
(1.8)
where we have taken the velocities of equation 1.7 to be orthogonal and equal to ±vF and neglected v0 as it only introduces a tilting or asymmetry to the conical dispersion, but does not affect any of the topological properties. The similarity of Weyl equation 1.8 to the Hamiltonian for graphene, H(q) = vF (qx σx + qy σy ), is obvious; however, 18
the use of the σz matrix in the Weyl equation means that no remaining terms can gap out the Weyl node. In graphene, the addition of a term proportional to m σz , such as from the breaking of sublattice symmetry (e.g., boron nitride), introduces a gap to the Dirac spectrum; whereas, such a term for the Weyl Hamiltonian merely shifts the Weyl node from k0 to k00 = k0 + (0, 0, m), rather than gapping it. Thus the Weyl node is robust to arbitrary perturbations that preserve the original symmetries and can only be annihilated by coupling to another Weyl node of the opposite chirality, a topological feature we describe below.
a)
b)
3 2 1
En
0
+ Weyl Pt
1 2 3
c)
4
0
2
2
4
3 2 1
En
0
- Weyl Pt
1 2 3 4
2
0
k‖B
2
4
Figure 1.7: a) Schematic band structure showing the linear dispersion around two 3D Dirac points. In a magnetic field, each Dirac point is split into two Weyl points. b) The Landau level spectrum for Weyl points of chirality ±χ, represented as a monopole or anti-monopole of the Berry flux. Notice that the zeroth Landau level is chiral, disperses only in one direction (i.e. the zeroth Landau level for a Weyl node of chirality χ has dispersion −χvF kz , where kz is parallel to the field.) The topological index of a Weyl node, called the chirality χ, can be taken as the handedness of the velocities in equation 1.7
χ = sgn[v1 · (v2 × v3 )] = sgn[det(vij )] = ±1 19
(1.9)
where vij is the j th component of vi [35]. Perhaps even more elegantly, the chirality χ can be equivalently computed in terms of the Berry flux F(k) over a two-dimensional surface S enclosing the Weyl node in the Brillouin zone 1 χ= 2π
I F(k) · dS(k) = ±1,
(1.10)
S
which establishes the Weyl nodes as monopoles of the Berry flux and χ as the Berry ‘magnetic’ charge. As pointed out by Nielsen and Ninomiya [36], the sum of the chiralities for all the Weyl points must be zero due to the periodicity of the Brillouin zone; thus, there must be an even number of Weyl nodes in the Brillouin zone, with half of each chirality. Inversion symmetry then implies that if a Weyl node χ exists at k0 , then one of opposite chirality −χ must exist at −k0 . Simultaneously, time reversal symmetry implies that for a Weyl node χ at k0 , a Weyl node of the same chirality must exist at −k0 .4 Thus if both inversion and time reversal symmetries are intact, Weyl nodes of opposite chiralities must exist at k0 (and −k0 ) and may annihilate one another, explaining why isolated Weyl nodes require breaking either inversion or time reversal symmetry. The conservation of zero total chirality leads to topologically-protected surface states, even in the gapless system of the Weyl semimetal [34]. As first proposed in the pyrochlore irridates, these surface states join the projections of two Weyl nodes of opposite chirality onto the surface Brillouin zone for a particular crystal face, thereby extending the two point semimetal Fermi surface into a single ‘arc’ (i.e., like a Dirac string joining a monopole with an anti-monopole). These surface states are guaranteed to exist at the Fermi level by correspondence to the edge states of quantum hall systems and can disperse in energy away from the Fermi level, but must exist in portions of the surface Brillouin zone non-overlapping with the projected bulk band 4 This is true since in equation 1.8 the momentum q is odd under inversion, odd under time reversal, while the Pauli matrices σi transform as a pseudovector (even under inversion, odd under time reversal).
20
structure. The disjointed ‘arc’-like nature of the Fermi surface can be rationalized by realizing that as the surface state arc approaches either Weyl node, its spatial character extends further into the bulk via hybridization with the bulk state at the Weyl node, and finally it reappears on the other surface, where it can connect to another Fermi arc, before re-entering the bulk again through the other Weyl node and completing the Fermi surface. A second novel aspect due to the topology of the Weyl semimetal is the presence of the chiral anomaly [35, 32]. The Landau levels for a single Weyl node in a magnetic field parallel to the z-direction are E0 = −χ~vF qz p En = vf sgn(n) 2~|n|eB + (~qz )2 , n = ±1, ±2, ...
(1.11)
where critically, the zeroth Landau level (ZLL) disperses in only one direction depending on the chirality χ of the Weyl node, as is shown in Fig. 1.7.5 It is useful to keep in mind that the equations we have presented are only the low energy expansions close to the Weyl point, and in real materials, Weyl nodes must eventually merge in a Lifshitz transition at some higher energy scale (since there must be an even number of them). Hence, the dispersion of the ZLL of a positive chirality Weyl node will link to the dispersion of the ZLL of a negative chirality Weyl node at some energy scale, somewhere in the Brillouin zone. In this case then, when a electric field E is applied parallel to the magnetic field B, the electronic states will drift under semi-classical theory as
~k˙ = −eE 5
(1.12)
A full derivation is very similar to the Landau levels of graphene, with only the addition of a qz σz term which is unaffected by the magnetic vector potential. Therefore, the Hamiltonian in harmonic √ q 2/l a 0 oscillator language is H = √2/lz a† −q b and the zeroth Landau level |n=0i has eigenvalue −qz . b
z
21
thus depopulating one Weyl node and populating the one of opposite chirality. Thus the electric field establishes an imbalance of the chiral charge, effectively charging a chiral battery at a rate proportional to E·B. To see this, let us define the chiral charge Q = e(Nχ − N−χ ), where Nχ is the number of uncompensated states of chirality χ. In a time dt, the Fermi momentum of one Weyl node increases by eEdt/~ by equation 1.12, while the Fermi momentum of the other Weyl node is reduced by the same amount. The density of states along the one dimensional ZLL is LB /2π, while the degeneracy of the ZLL (how many copies of the zeroth chiral Landau band we have) is A⊥ B/Φ0 = eA⊥ B/2π~. Multiplying these degeneracies, we obtain for the rate of chiral charge accumulation dQ eE LB eA⊥ B = 2e ∗ ∗ ∗ dt ~ 2π 2π~ e3 V E · B = 2π 2 ~2
(1.13)
where we have taken V = A⊥ LB to be the system volume. This chiral imbalance can manifest itself through a negative magnetoresistance6 and other exotic magnetotransport signatures, such as the anomalous hall effect and the chiral magnetic effect, where a pure magnetic field can induce a non-equilibrium electric current. Moreover, proposals have been put forth to utilize its nonlocal electronic transport properties or sensitivity to magnetic fields for practical applications [38]. This thesis describes Landau quantization and quasiparticle interference measurements performed by Jeon and Zhou on the Weyl semimetal candidate, the ultra-high mobility II-V semiconductor Cd3 As2 [39]. At zero magnetic field, Cd3 As2 is actually a Dirac semimetal since inversion and time reversal symmetries are preserved. The band touching points are formed at the crossing of two doubly-degenerate bands and thus represent two overlapping Weyl points. As we have mentioned, in this case, the 6
Although in real materials, such as Cd3 As2 , this may be masked by positive contributions to the magnetoresistance [37].
22
Dirac point is generally susceptible to gapping, so an additional symmetry is required to preserve the gapless Dirac points in Cd3 As2 . As discovered in Refs. [40, 41], if the three-dimensional Dirac points occur along certain high symmetry directions in the Brillouin zone, they are protected by crystalline space group symmetries. For example, the C4 screw symmetry around the kz axis in Cd3 As2 protects the two Dirac nodes in this direction. Hence, Cd3 As2 can host a Weyl semimetal phase when time reversal symmetry is broken through a magnetic field applied along the c-axis ([001] direction), which still preserves the original rotational symmetry. However, due to the (112) plane being the natural growth and cleavage plane of Cd3 As2 , we could not access the Weyl semimetal phase by our experimental restriction of applying a magnetic field perpendicular to the cleaved sample surface. Nevertheless, our measurements confirm many aspects of the Dirac semimetal phase, such as the extended linear dispersion away from the Dirac points and the expected two-fold conduction band degeneracy, resolvable at high magnetic fields, and reveal for the first time the microscopic details and band structure regime relevant to the ultra-high mobility seen at the Fermi level [37].
1.5
Thesis Outline
This thesis first begins with Chapter 2 which provides a short exposition of the technique of scanning tunneling microscopy and a discussion of the quasiparticle interference technique, the method by which momentum space information can be extracted from a real space STM measurement. Next in Chapter 3, we describe the design, operation, and performance of the novel dilution refrigerator-based STM constructed during the course of this thesis and used exclusively for the experiments presented herein. The first experiment performed on this instrument, described in Chapter 4, investigated the unconventional superconducting state in the heavy fermion CeCoIn5 23
at milli-Kelvin temperatures and proved definitively that the order parameter is of dx2 −y2 symmetry. In addition, this experiment demonstrated the onset of a spectroscopic pseudogap prior to superconductivity in the strongly-hybridized heavy fermion band. In Chapter, 5, we leverage the high field capabilities of the machine to dissect the intriguing band structure of Cd3 As2 and verify its predicted 3D Dirac dispersion, which unexpectedly survives to higher energies than originally believed. Here, Landau level spectroscopy is extended to a three-dimensional band-structure, in distinction to its usual application in STM to two-dimensional systems, such as surface states and graphene. We conclude in Chapter 6 with a discussion of future technical improvements and potential extensions to the experiments performed. The appendices provide additional information on various aspects, including the day-to-day operation of the system, an alternative conductance mapping technique used for the experiments reported in the thesis, and technical details of the simulations used to understand the data. Moreover, we briefly discuss preliminary data on the search for enhanced charge order in underdoped Bi2 Sr2 CaCu2 O8+δ at high magnetic fields. It is hoped that others may benefit from the lessons learned and may use this thesis to push the limits of STM to even colder temperatures, stronger fields, and quieter performance to behold the marvel of material properties at the atomic scale and at the frontiers of phase space.
24
Chapter 2 The Basics of Scanning Tunneling Microscopy Microscopy, from the Greek words micros, meaning small, and skopos, meaning to observe, is the development of tools to view and study objects unresolved by the human eye. Among the many techniques in this field, scanning tunneling microscopy is perhaps the most powerful, as its resolution is set not by the wavelength of a probe beam of photons or electrons which scatter from a sample, but by the overlap between the quantum mechanical wavefunctions of the atoms on a probe and sample. In this chapter, we introduce the theoretical foundations for this technique and highlight the method of quasiparticle interference imaging, which can provide momentum space information from an intrinsically real space measurement.
2.1
Theory
A complete history of STM must start with the demonstration of quantum tunneling by Leo Esaki in semiconductor tunnel diodes (1957) [42] and by Ivar Giaever in superconductor-insulator-superconductor junctions (1960) [43]. These seminal experiments revealed the quantum nature of matter at reduced length scales, showing that 25
electrons could pass through a classically forbidden region, if that region, or barrier, is made thin enough. Indeed, the STM is merely an inspired application of this idea, by making one of the tunneling contacts a sharp metallic wire, called the “tip”, fully positionable over the the other tunneling contact, the sample, with vacuum as the energy barrier in between. Using piezo-electric motors and tube scanners that enable three dimensional motion with sub-picometer accuracy, the tip can be placed within ten angstroms from a clean, conductive sample surface, reducing the barrier length until the tunneling current can be measured as a function of the lateral position of the tip, as it is scanned across the sample surface. Mathematically, the sample electron wavefunctions ψs decay across the vacuum barrier as [44]
ψs (z) = ψs (0)e−κz
(2.1)
where for electrons near the Fermi level Ef , √ κ=
2mφ ~
(2.2)
with φ denoting the work function of the material. The tunneling amplitude, proportional to |ψs (zt )|2 , is exponentially sensitive to the position of the tip zt above the sample surface, and this extraordinary sensitivity underlies the atomic resolution of the STM. For metals, φ ∼ 5 eV and the change in tunneling current is an order of magnitude for only a single ˚ A of tip displacement. The total tunneling current for a bias voltage −V applied to the sample can be quantitatively calculated via Fermi’s golden rule of time-dependent perturbation theory. As shown schematically in Fig. 2.1, the density of states (DOS) and occupation level of the sample EFsample are shifted rigidly up by eV in energy with respect to tip density of states and occupation level EFtip . It is important to distinguish that the 26
‘
‘
‘
Figure 2.1: Representation of tunneling from sample to tip at −V bias to sample. Conservation of energy implies filled electrons states of the sample tunnel horizontally across to equal energy, empty states of the tip. Reproduced from [45].
action of the bias −V is not to fill additional levels in the sample DOS, but rather to raise the energy of the originally occupied states with respect to the tip’s Fermi level such that elastic tunnelling out of the occupied sample states to the corresponding empty tip states can occur. With this understanding, the correct equations can be written down, accounting for both the dominant current tunneling from sample to tip and (for completeness) the much smaller, reverse current from tip to sample 4πe I(V ) = − ~
Z
∞
(2.3) f (EF − eV + )(1 − f (EF + )) {z } | sample to tip − (1 − f (EF − eV + ))f (EF + ) ρs (EF − eV + )ρt (EF + )|M |2 d | {z } tip to sample Z 4πe ∞ =− f (EF − eV + ) − f (EF + ) ρs (EF − eV + )ρt (EF + )|M |2 d ~ −∞ −∞
(2.4) F where f () = (1 + exp( −E ))−1 is the Fermi function with the original chemical kB T
potential EF (of the tip and sample when V = 0), and ρs () and ρt () denote the sample and tip DOS, respectively. The matrix element |M |2 captures the square amplitude of the overlap between tip and sample wavefunctions, including their spatial 27
character and decay rate κ across the barrier. Assuming ρt and |M |2 independent of energy and kB T smaller than features of interest so that Fermi functions are step-like
f (EF − eV + ) − f (EF + ) =
0 for < 0 1 for 0 < < eV 0 for eV <
the expression can be simplified to Z 0 4πe 2 I(V ) = − |M | ρt ρs (EF + 0 ) d0 ~ −eV dI 4πe (V ) = − |M |2 ρt · ρs (Ef − eV ), dV ~
(2.5)
where we have redefined 0 = − eV .1 Thus, in theory by performing current-bias sweeps at a point, the differential conductance
dI dV
can be numerically computed to
reveal the energy-resolved local DOS of the sample, the key experimental signature of the underlying physics, from the supeconducting energy gap to the Kondo resonance. Operationally, a fixed position for the tip zt is first chosen via a setpoint condition I0 (V0 ), the feedback loop is then opened to maintain zt as the bias voltage is swept, and a small AC modulation is summed on top of the swept voltage such that a lock-in measurement can be performed to detect the differential current at the modulation frequency. In addition to local point spectroscopy, the tip may be scanned across the surface to add the two lateral real space degrees of freedom x and y to any measurement. This simplest scanned measurement is called topographic mode, where a feedback loop adjusts the height of the tip z as it moves across the surface to keep the total 1
The sign convention for these equations is such that +V tunnels electrons out of the sample. Generally in STM, one applies bias to the sample, such that experimentally a negative voltage −V is applied to the sample to tunnel electrons out of the sample. For this setup, we generally just remember that −V probes filled states of sample, +V probes empty states of sample.
28
Point Spectra - LDOS
Topographic Mode
Spectroscopic Imaging
dI/dV (nS)
0.8 0.6 0.4 0.2 0 −150 −100
−50
0 50 Bias (mV)
100
150
Figure 2.2: The three modes of STM operation exemplified on the high-Tc superconductor Bi2 Sr2 CaCu2 O8+δ , perhaps the single one material that proved the power of STM as a condensed matter probe. Local dI/dV measurements reveal the unconventional superconducting gap spectrum (contrast with s-wave superconducting gap in Fig. 3.5). Constant current topographic (z) image of the BiO plane shows the individual Bi atoms and a stripe-like bulk supermodulation. Finally, by plotting the differential conductance at a particular voltage (dI/dV (V = 22 mV )) over the real space view, the STM directly visualizes the ordering of the electronic states into a ‘checkerboard’ pattern.
tunneling current constant. Topographic images z(x, y) often reveal the underlying structural features, but in theory is also sensitive to the local conductivity. However, the most powerful STM measurement is called spectroscopic imaging. Here, it is differential conductance measurement the resulting maps of
dI dV
dI dV
that is performed on a grid of points, and
(x, y, V ) ≡ C(x, y, V ) at a particular energy V reveal the
inhomogeneity of electronic states, such as the ‘checkerboard’ charge ordering in the high-Tc superconductor Bi2 Sr2 CaCu2 O8+δ as shown in Fig. 2.2, and be analyzed for quasiparticle interference, as described below. Such conductance maps often take several days to complete; hence, it is the ultimate test of the stability of the STM, to both long time scale drifts and instantaneous vibration performance.
2.2
Quasiparticle Interference Imaging
Electronic inhomogeneity on metal surfaces can arise from many sources, from random fluctuations due to chemical doping or periodic order due to charge density waves. 29
Perhaps of the most generic source is the effect known as quasiparticle interference (QPI), whereby the breaking of translational symmetry due to defects on the surface of a crystal, such as an atomic step edge or localized point disorder, mixes the Bloch states of the translationally invariant system and introduces (Friedel) oscillations in the local charge density. When the plane wave state Ψ1 = eik1 ·r interferes with a second plane wave state Ψ2 = eik2 ·r due to a scattering potential, the resulting charge density
ρ ∝ |eik1 ·r + eik2 ·r |2 = 2(1 + cos(q · r))
(2.6)
acquires a modulation at the wavevector q = k1 − k2 , with an amplitude that decays away from the scattering center. A STM conductance map C(x, y, V ) visualizes these modulations of the charge density, ideally over a large field of view where many defects contribute to many quasi-independent modulations.2 The wavelength and direction of the modulations can then be determined by taking the two-dimensional Fourier transform of this map, once the length scale of the piezo scanner is calibrated from the atomic Bragg peaks. A typical set of QPI data is shown for the prototypical example of “surface waves” on Cu(111) in Fig. 2.3. To discuss how the wavevectors q relate to the Fermi surface of the underlying material, we note that scattering processes conserve energy so that the set of possible q vectors span the vectors that connect two points on the Fermi surface. For now let us assume that the Fermi surface is two-dimensional, such as in the case for surface states of Cu(111) or topological insulators, as the three-dimensional Fermi surface requires additional assumptions. Then heuristically, we might expect the Fourier transform ˆ x , qy , V ) = F{C(x, y, V )} to be proportional to the auto-convolution of the map C(q 2
Experimentally, the range L of the real space field of view determines the momentum q space resolution 2π/L of the Fourier transform map; while the real space resolution L/N, where N is the number of pixels of the map, determines q space range ±πN/L.
30
Energy
kf
Q
ki
100 Å
ky
kx
(1.0, 1.0) Å-1 -220 mV
-20 mV
180 mV
Figure 2.3: The surface state of Cu(111) displays quasiparticle interference caused from the scattering from carbon monoxide molecules and step edges on the surface. The energy-resolved spectroscopic maps display wave modulations, whose wavelength decreases with increasing energy (the overlaid triangle shows that the real space scale does not change between images) The 2D Fourier transform of such real images reveal a ring of wavevectors whose radius Q is equal to twice the k vector of the contour of constant energy of the parabolic surface state band structure, shown in the left schematic. Q = 2k reflects the relative importance of backscattering, which in the limit of delta function potentials and infinite lifetime should be the only wavevector existing [46, 47].
Fermi surface intensity I(k) in two-dimensions
ˆ V)∝ C(q,
Z
I(k, V )I(k + q, V )d2 k.
(2.7)
BZ
This is the so-called joint density of states (JDOS) approximation.3 The crucial insight is that the Fourier transform of a real space image, due to interference modulations, can be used to extract momentum space information about the Fermi surface shape, although generally the convolution is not exactly invertible, especially in the presence of noise. However, for many simple Fermi surface geometries, such as a 3
The JDOS approximation is a model for QPI and should not be treated as an exact equation. JDOS in general underestimates the intensity of 2kF backscattering, which should dominate hard wall and delta function potentials.
31
circle or a square, the auto-convolution of the Fermi surface, measured by the QPI technique, remains dominated by the same shape (i.e., square or circle) except for the scaling q = 2 k. This is most clearly demonstrated in Fig. 2.3 for the circular Fermi surface contours of the parabolic surface state band of Cu(111). More generically, we can define a matrix element that modulates the scattering contribution depending on the initial and final states such that the equation is modified to
ˆ V)∝ C(q,
Z
I(k, V )T (k, q)I(k + q, V )d2 k.
(2.8)
BZ
Such a matrix element might arise due to the spin projection of the initial and final states of spin-momentum locked surface states in topological insulators or due to superconducting coherence factors for quasiparticles in d-wave superconductors. The next question arises as to what to use for the intensity of the Fermi surface I(k). Generally, we can use the experimental ARPES intensity for I(k) when it can be shared with us by our ARPES collaborators. However, when ARPES data is not available, such as for the heavy fermion experiments described in this thesis, I(k) can be taken as the Green’s function for some model band dispersion (k)
ˆ ω) = I(k, ω) ∝ G(k,
1 , ω + iΓ + (k)
(2.9)
where we now imagine trivially computing the JDOS at arbitrary energy ω (i.e., bias voltage V), and the lifetime Γ can be chosen to suitably broaden the resulting features in accordance with experimental data. When the Green’s function is used for I(k), equation 2.7 is precisely the first term of the Born scattering series for an impurity.
32
These equations are operationally computed via matrix Fourier transforms that take advantage of the convolution theorem.4 We extend the discussion to QPI in the case of three dimensional band structures. Formally, since the Fermi surface must be considered in the three dimensional Brillouin zone and the STM can only measure the component of the modulation projected onto the surface qk , the joint density of states becomes the integral over 3D Fermi surface and over the qz component of the scattering vector Z Z
ˆ k, V ) ∝ C(q
I(k, V )I(k + q, V )d3 k dqz ,
(2.10)
3D BZ
which is significantly more difficult to compute because of the added dimensionality. Accordingly, the simplifying assumption one makes is that the QPI signal for a 3D band structure is a sum, perhaps weighted sum, of the individual 2D QPI from slices of the Fermi surface for fixed kz , where z is perpendicular to the surface. ˆ k, V ) ∝ C(q
X kz
Z
I(kk , kz )I(kk + qk , kz )d2 kk .
w(kz )
(2.11)
2D BZ for fixed kz
This less general equation restricts q to have only zero qz component, which is approximately correct in limit of strong backscattering for a Fermi surface with symmetry about the kz = 0 plane. In general, we can even further simplify by saying the the sum in 2.11 is dominated by one or two particular values of kz , such as kz = 0 or kz = π as in the case of the CeCoIn5 measurements to be discussed in this thesis. Despite some of these inconveniences, one could wonder why QPI as a technique would be favored over conventional ARPES which can measure the momentum struc4
In general, to take advantage of the convolution theorem, the integrand in equation 2.8 must be factorized into the product of two terms, one involving k and one involving k + q. For example, T (k, q) in the case of spin projection takes the form cos (θ1 − θ2 )/2 , where θ1 (k) and θ2 (k + q) are the orientations of the spinor at the initial and final states. This is computed by expanding the expression into 1/2 ∗ (1 + cos(θ1 ) cos(θ2 ) + sin(θ1 ) sin(θ2 )) and thereafter summing the three separate convolutions by FFT.
33
ture of materials directly and quickly. Certainly in a direct head-to-head match, ARPES is strongly favored to QPI. However, QPI is applicable in many situations when ARPES is not, such as above the Fermi level, in a magnetic field, and when high energies resolution is needed. This is particularly true for the rapid dispersions of heavy fermion materials, where the current thesis used ∼ 80 uV energy resolutions to resolve the “flat” bands that emerge within millivolts of the Fermi level. Finally, we argue that sometimes it useful to measure scattering (the effect) directly, rather than just the Fermi surface (the cause), because it is ultimately scattering than determines the transport properties of the material. This is beautifully illustrated in the experiments on topological insulators by Roushan [48], which directly demonstrated the absence of backscattering, the key consequence of spin-momentum locking in those materials.
34
Chapter 3 The Dilution Refrigerator Scanning Tunneling Microscope This chapter is based upon the publication: Misra S., Zhou B. B. et al.,“Design and performance of an ultra-high vacuum scanning tunneling microscope operating at dilution refrigerator temperatures and high magnetic field.” Review of Scientific Instruments 84, 103903 (2013) [10].
We describe the construction and performance of a scanning tunneling microscope (STM) capable of taking maps of the tunneling density of states with sub-atomic spatial resolution at dilution refrigerator temperatures and high (14 T) magnetic fields [10]. The fully ultra-high vacuum system features visual access to a two-sample microscope stage at the end of a bottom-loading dilution refrigerator, which facilitates the transfer of in situ prepared tips and samples. The two-sample stage enables location of the best area of the sample under study and extends the experiment lifetime. The successful thermal anchoring of the microscope, described in detail, is confirmed through a base temperature reading of 20 mK, along with a measured electron temperature of 240 mK. Atomically-resolved images, along with complementary vibration 35
measurements, are presented to confirm the effectiveness of the vibration isolation scheme in this instrument. Finally, we demonstrate that the microscope is capable of the same level of performance as typical machines with more modest refrigeration by measuring spectroscopic maps at base temperature both at zero field and in an applied magnetic field.
3.1
Introduction
Scanning tunneling microscopy (STM), since its development almost 30 years ago, has become a powerful technique in condensed matter physics, providing not only structural information about surfaces, but also spectroscopic measurements of the electronic density of states at the atomic length scale. However, most instruments operate at temperatures above 1 K, limiting access to exotic electronic phases and quantum effects expected at lower temperatures, which are studied as a matter of routine by other techniques. Generally, very little spectroscopic information about the electronic density of states is known at dilution refrigerator temperatures, usually being limited to what can be learned using either point contact spectroscopy or planar tunnel junctions. Moreover, STM can make such measurements on the atomic length scale, allowing it to probe systems, such as single spins and atomic chains, which are not directly accessible any other way. While the integration of STM with a dilution refrigerator can be conceptually reduced to simply attaching the microscope to the end of a mixing chamber in lieu of some other cryogenic refrigerator, the technical requirements for sub-Angstrom positioning of an STM tip above an atomically clean surface are often at odds with those for cooling a sample to milli-Kelvin temperatures. For example, when attaching the microscope to the refrigerator, the former would favor the use of a soft mechanical joint using springs, which would isolate vibrations, while the latter would favor the 36
use of a rigid mechanical joint with a metal rod, which would provide a strong thermal contact. Nevertheless, a number of STM instruments have been developed that cool the sample using a dilution refrigerator [49, 50, 51, 52, 9, 53, 54, 55]. However, among these, few feature ultra-high vacuum (UHV) environments, which would facilitate the in situ preparation of tips and samples, a crucial step in preparing many samples and functionalizing STM tips [55, 51, 9]. Moreover, few have the level of stability and performance required to measure spectroscopic maps of the electronic density of states with atomic spatial resolution, crucial to obtaining detailed information about the electronic state of the compound under study [50, 54, 51, 9]. Here, we describe the construction and performance of a home-built STM designed specifically to extend the level of functionality and stability common in higher temperature systems to dilution refrigerator temperatures.
3.2
Ultra-High Vacuum Assembly
The successful integration of a dilution refrigerator into an ultra-high vacuum environment has the unmeasurable benefit for scanning tunneling microscopy that the full range of samples available to the technique could be studied, and standard techniques for the in situ preparation of tips and samples could be used without alteration. Although standard dilution refrigerators contain materials, such as nylon, brass and soft solder, which are anathema to ultra high vacuum, substitution by UHV-compatible materials (PEEK, OFHC copper, and high temperature solder) and the adoption of proper cleaning methods has been successfully implemented in a number of systems [56, 57, 58]. The remaining difficulty lies in devising a scheme to transfer tips and samples between various UHV stages and the microscope. Toward this end, we mount a specially designed bottom-loading dilution refrigerator insert onto a z-manipulator which can translate the insert and attached microscope 37
a
d
Insert Manipulator
b
Manipulator Platform & Track
Cryostat
Manipulator Bellows
OVC
3.3 c m
e Measurement Position Sample Access Position
Insert Head Flange
IVC Bellows
Chambers 2.16 m
Helium Space
c
Preparation Chamber
UHV Space IVC Bottom Flange Cryostat Bottom Flange
Transfer Chamber
2.71 m
Figure 3.1: (a) This 3D CAD drawing shows a zoomed-in cross-section of the insert, manipulator and cryostat, whose connection to the UHV chambers is shown in (b). When the manipulator platform moves down its track (yellow), the manipulator and IVC bellows (red) contract, and the insert (brown) moves out the end of a UHV neck at the bottom of the cryostat into the chambers below, translating the attached microscope between the measurement and sample access positions labeled in (b). The relevant Conflat flanges which interface the insert to the IVC (IVC head flange), the IVC to the cryostat (IVC bottom flange), and the cryostat to the vacuum chambers (cryostat bottom flange) are shown in pink. (c) The UHV chambers beneath the cryostat include a transfer chamber, a preparation chamber, and a load lock (green). The wobble sticks and manipulators used to transfer samples are shown in red. The focal point of the transfer chamber can accommodate a number of evaporation sources (two are shown in purple). The left-hand focal point of the preparation chamber has a resistive heater stage for samples or tips (not shown), and can have up to four evaporators pointed at it (two are shown in cyan). The right-hand focal point of the preparation chamber has an e-beam heater stage with an Ar ion sputter gun pointed at it (dark blue). (d) This drawing shows a close-up of the e-beam heater stage, with the sample holder shown in green. Alumina pieces are shown in light blue, and the filament in red. (e) Photograph of the transfer and preparation chambers attached to the cryostat. 38
between the measurement and sample access positions within contiguous UHV space (Fig. 3.1b). This UHV space of the instrument extends up from the chambers via an inner vacuum can (IVC) with a flexible bellow up to a head flange on the insert (Fig. 3.1a). With the exception of the wiring interface, which is through a chamber at the top of the insert and connects to the head flange through a series of tubes, the head flange is the terminus of the UHV space. The top of the rigid insert is secured to a heavy duty (non-UHV) z-manipulator. This manipulator lowers the entire insert down 65 cm, collapsing the IVC bellows. This, in turn, moves the microscope, which normally sits at the center of a 14 T magnet (103 mm bore diameter) when the manipulator is up, through the bottom neck of the cryostat, and produces it at the center of the UHV chamber below. After opening a rotary door on the radiation shield of the refrigerator using a multi-motion wobble stick, we have direct visual access to the microscope itself. The UHV utilities in the three chambers attached to the cryostat have been specifically designed to enable the implementation of the full suite of recipes for in situ preparation of spin-polarized STM tips and samples[59]. New tips and samples are introduced into the UHV chambers through a standard load lock attached to the preparation chamber (Fig. 3.1c). The preparation chamber contains two points which each lie at the focus of multiple ports of the vacuum chamber, one of which has an e-beam heater (Figure 3.1d) and sputter gun to clean tips and metal samples, and the other of which has a resistive heating stage and evaporators which can be used to grow thin metal films on them. To allow for evaporation onto a cold sample, the transfer position of the dilution refrigerator insert sits at the focal point of three ports of the transfer chamber, to which standard evaporators can also be attached. Finally, cleavable samples can be both cleaved and stored in the preparation chamber. The operating base pressure of the system is ∼ 10−10 torr. The UHV chambers can achieve this level of vacuum simply by baking to 130 C for two days. The insert 39
has a pair of heaters located near the microscope, but can only be baked to 60 C. Despite this limitation, after cooling to liquid helium temperatures, the insert does not change the level of vacuum in the transfer chamber, even when swapping samples or tips. As shown in the last section, this level of vacuum is more than sufficient to leave prepared surfaces clean for examination with the microscope.
3.3
Vibration Isolation
The quality of the data taken by any STM is largely determined by its ability to limit the level of vibrations in the tip-sample junction, to which the tunnel current is exponentially sensitive. For our microscope, the need for strong thermal coupling to the dilution refrigerator precludes the use of springs at the microscope itself, a common technique which is remarkably effective. Instead, as we outline in Figure 3.2, we have isolated the entire instrument shown in Figure 3.1b from external acoustic and floor-borne sources of noise. Vibrations present in the floor of the laboratory are first attenuated down to very low frequencies (∼ 1Hz) by a set of six passive pneumatic legs which float a 30 ton concrete plinth, as shown in Figure 3.3. A passively isolated 4 ton granite slab sits on top of this concrete plinth, providing an additional layer of isolation from floor-borne noise. To realize this low level of vibrations in the microscope itself, the instrument (cryostat and chambers) is secured to a dissipative heavy duty aluminum frame, whose only rigid attachment is to this vibration-isolated granite slab above. A similar two-tiered scheme was realized to attenuate acoustic vibrations. Two layers of acoustic shielding are realized by surrounding the plinth first by an acoustic enclosure, and then surrounding that enclosure by a second room built from grout-filled concrete blocks. This scheme to isolate vibrations can be rendered useless unless proper care is taken in handling the large pumps and pumping lines required to run a dilution 40
(7) +y
(6)
+z
59 cm
+x
(3) (8) (9)
(9)
(1) (4) (5)
(8) (2)
Figure 3.2: The instrument (1) is mounted to a custom-made aluminum support structure (2), both of which are surrounded by a radio frequency (RF) isolation enclosure (3). In order to facilitate the mating of the insert to the cryostat, which requires moving the chambers out of the way, the chambers are mounted to an aluminum table, which is attached to the aluminum frame by way of special kinematic mounts. This entire support structure is suspended freely by securely bolting to a 4 ton granite table (4) above. This table floats on a set of 6 passive isolators, and serves as one isolated stage. This, in turn, rests on a 30 ton concrete plinth (5), which itself sits on a set of another 6 heavy passive isolators, and defines a second isolated stage. An acoustic enclosure (6) surrounds the entire structure. The plinth, the acoustic enclosure, and an external grout-filled concrete wall (7) connect only through the floor of the basement lab. A concrete block (8) and gimbal (9) is used to isolate vibrations which might be transmitted by the pumping lines between the floor and the plinth, and again between the plinth and the granite table (pumping lines delineated by yellow line). (left inset) This shows a 3D CAD drawing of our double gimbal, whose purpose is to decouple the motion of the two flanges (green), which are each attached to rigid pipes. This is accomplished by attenuating the motion of either of the flanges with respect to the central elbow (pink) through the use of two sets of edge welded bellows (red). The position of this elbow is determined by vacuum forces balanced by the tension on wire ropes (black) connected to the arms (brown). If the pipe attached to the bottom flange of the assembly were to move in +z (+x), then the horizontal (vertical) arm would twist upwards (right) on an axial bearing (light blue). Similarly, if it were to move in +y, the base plate (purple) would twist to accommodate the motion. (right inset) Aerial photograph of the instrument with detachable RF room top moved aside. 41
3
Velocity SD (nm/(s ⋅√Hz))
10
Lab Floor Plinth Granite Noise Floor
2
10
1
10
0
10
−1
10
0
100
200 300 Frequency (Hz)
400
Figure 3.3: This plot shows the velocity spectral density, as measured by a Wilcoxen 731A, present on the lab floor (blue), on top of the plinth (red), and on top of the granite slab (green). The combined baseline sensitivity of the accelerometer and spectrum analyzer is shown as the noise floor (black). The data were taken while running the dilution refrigerator. refrigerator. The pumps generate a lot of noise, and the pumping lines not only transmit these vibrations, but their stiffness can mechanically short the pneumatic isolation stages together. To attenuate high frequency vibrations, the four gas lines, which includes the still and 1K pot pumping lines, the condenser and the cryostat exhaust, are cast in a 0.5 ton concrete block located on the lab floor (Fig. 3.2). To attenuate low frequency vibrations and prevent mechanical shorting of subsequent pneumatic stages, the 1K pot pumping line, the condenser and the dewar exhaust have long looped sections of formed bellows between the lab floor and the plinth. To achieve the same effect on the much larger diameter (and thus much stiffer) still pumping line, a double gimbal based on the design of Ref. [60] (left inset of Fig. 3.2, manufactured by Energy Beams, Inc.) is used to bridge the gap between the lab floor and the plinth. This combination of concrete block and gimbal is repeated again when going from the plinth to the granite table. As shown in Fig. 3.3, the end result √ of our isolation scheme establishes sub-nm/(s· Hz) vibration levels approaching the 42
noise floor of our accelerometer for a wide range of frequencies even with all pumps attached and running. Having created a suitably low vibration environment, the STM, which we describe in the next section, must be made as rigid as possible to realize the level of performance required to take low noise measurements, which we describe in the last section.
3.4
The Dilution Refrigerator and Microscope Head
The last challenge is the conceptually simple step of attaching an STM head to a dilution refrigerator, with the goal of attaining the lowest possible temperature while retaining the maximum amount of functionality in the STM head. Thermally, the insert on our system is a fairly standard design, but with four notable exceptions (Figure 3.4a) to accommodate the UHV compatibility of the system. First, given that the IVC is a UHV space, exchange gas cannot be used to cool the insert from high temperatures down to 4 K, either for the initial cooldown of the system, or when transferring samples. Instead, we have an oval-shaped mechanical heat switch, operated using a rotary feedthrough, which allows the mixing chamber to be thermally shorted to the 1K shield via mechanical contact. This allows us to cool the system from room temperature to 2 K in around 40 hours during the initial cooldown, and from around 40 K to 2 K in around 6 hours when transferring samples. Second, because the insert needs to be moved up and down and the 1K pot has a fixed length pickup tube, about half the helium in the main bath is accessible to the pot, resulting in shortened time between refills of the main bath. That the cryostat has two satellite necks (to accommodate the demountable magnet current lead and cryogenic services) exacerbates this by increasing the helium consumption. Still, the helium consumption rate with the dilution refrigerator running is around 18 liters a day, resulting in a hold 43
Microscope
Shutters
69 cm
e
2.5 cm
f
13 cm
Filters Silver Rods
Washboards
136 cm 1 K Shield
Rotary Door
d
2.0 cm 56 cm
Tail
Wiring Interface
DR Strike Plate
c
3.3 cm
Accessible by 1K pot
1K Pot
Mixing Chamber
b
a
3.7 cm
3.5 cm 2.0 cm
g
2.5 cm
Figure 3.4: (a) This 3D CAD cross-section of the general assembly highlights the non-standard features of the insert. The oval cam and strike plate used to pre-cool the microscope are shown in yellow. The approximate volume of the main bath accessible to the 1K pot pickup line is shown in purple. The rotary shutter on the 1K radiation shield is shown in cyan. The two sets of radiation baffles on the cryostat, which sit at roughly 77 K and 4K, are shown in green. The location of the 1K pot (purple), the 1K radiation shield, the tailpiece (tail), and dilution refrigerator (DR) are also labeled. (b) This is a 3D CAD drawing of STM tailpiece, highlighting the washboards (cyan), the tip and sample RF filters (purple), and the silver rods (yellow). (c) This shows a 3D CAD drawing of the microscope, with the z-motor shown in blue, the x-motor shown in red, and the sample cubby shown in green. A radiation shield which shields the tip and the sample is hidden from view. (d) A cross-section of the z-motor is shown here, with the prism in purple, the scan piezo in red, piezo stacks in blue, the bottom plug in dark green, and the capacitance sensor in light green. (e) A cross-section of the x-motor, with the prism in purple, the piezo stacks in blue, the arms which hold the sample cubby in dark green and the capacitance sensor plates in light green. (f) This is a 3D CAD model of the sample cubby, showing the samples (green) and the PEEK lid (white). This lid, when pushed down, compresses two BeCu springs on the backs of the samples. The two dovetail pieces (red) ride on tracks, and can lock the lid into place. (g) Photograph of the microscope with the milli-Kelvin radiation shield pulled down to reveal access to the empty tip receptacle and sample cubby (the CAD in (f) shows a newer version of the cubby than the photograph does).
44
time for the main bath of 4 days. Finally, in order to have access to the microscope inside UHV, the radiation shields in the cryostat and the 1K shield need to be able to be opened inside the vacuum space, which could compromise their performance. These appear not to introduce any unexpected radiative heat load, as the dilution refrigerator was measured to achieve a base temperature of 8 mK (measured using a cobalt-60 nuclear orientation thermometer) and had a cooling power of 400 µW at 100 mK with only the thermometry installed. To cool the microscope to milli-Kelvin temperatures, we borrow standard techniques used for sample-in-vacuum dilution refrigerator instruments (Figure 3.4b). The primary heat load added when installing an instrument on the mixing chamber comes from the wiring. Both the shielded and unshielded lines extending down from room temperature are thermally anchored first at 4K, then at the 1K pot using 3 cm long washboards, and then connected to either flexible stainless/ NbTi coaxial cable (custom from New England Wire Technology) or NbTi wire. Because NbTi superconducts at 1K pot temperatures, these effectively act as a thermal break between the 1K pot and the mixing chamber. Between the mixing chamber and the microscope, however, we would like to maximize the thermal conductivity, and accordingly we use silver-coated copper coaxial lines and wires that are anchored using 10 cm long washboards screwed tightly into a copper stub that is attached to the mixing chamber. In addition, the tip and bias lines are fed through a lumped element RF filter (VLFX-1050+ from Mini Circuits) at the mixing chamber to filter out unwanted high frequency noise in the tunnel junction. To efficiently cool the body of the microscope itself, which is 42 cm away from the mixing chamber, we link the two with silver rods (3N5 purity, 50mm2 in cross-section, from ESPI Metals, Inc.) which have been annealed [61] to enhance their thermal conductivity. These fit in a PEEK frame which houses electrical connectors and serves as a secure mechanical attachment point for the microscope. With the microscope installed, the base temperature of both the 45
mixing chamber and the microscope was measured to be 20 mK using a ruthenium oxide thermometer anchored at the microscope. The mixing chamber now retains 260 µW of cooling power at 100 mK, leaving sufficient flexibility for adding more lines to the system when more elaborate experiments need to be done. The microscope (Fig. 3.4c) contains three functional blocks: a z-axis piezo motor, an x-axis piezo motor, and a two-sample cubby. Both the motors are Pan walkers [62], in which a triangular sapphire prism (custom, from Swiss Jewel Company) can be moved along a single axis of motion using three pairs of piezoelectric shear stacks (Model P121.01H from Physik Instrumente L. P.) in a slip-stick motion. The bodies of both the motors are made of OFHC copper, coin silver, and PEEK pieces held together using non-magnetic silicon bronze screws (custom, from Swiss Screw Products, Inc.). The z-motor (Fig.
3.4d), which provides a total of 4 mm of vertical
motion, is used to approach and withdraw a scan piezo (3 ˚ A/V sensitivity in z, 9.5 ˚ A/V sensitivity in x/y, EBL #4 material from EBL Products, Inc.) which is glued to the inside of a cylindrical cavity in the prism. This motor reliably produces 80 nm-sized approach steps at low temperatures with a drive voltage of 375V supplied by a Nanonis PMD4 piezo motor driver. Its absolute position can be tracked using a cylindrical capacitor formed by a metal end-plug in the prism and a corresponding piece which is part of the (static) microscope body. The x-motor provides a total of 7 mm of horizontal motion in 280 nm-sized steps to the attached sample cubby (Fig. 3.4e). A pair of capacitors, each formed by the arms that carry the sample cubby and two internal (static) plates on the microscope body, allow us to track the absolute horizontal position of the sample. The x-motor has been designed to have such a large offset range specifically to allow us to move the tip between two samples (Fig. 3.4f) in the cubby. In addition to being designed to accommodate two samples, the cubby has a PEEK lid, which, when opened using the wobble stick, allows samples to be easily slid into the stage, and, when closed using the wobble stick, compresses two 46
BeCu spring contacts firmly into the samples and locks into place. This mechanism provides a more solid mechanical and thermal joint between the sample holders and the sample cubby than would otherwise be possible. In addition to the two samples, the STM tip can also be swapped in situ by plugging it with a wobble stick into a BeCu collet. While the ability to offset the sample over a large range provides considerable conveniences as discussed below, the lowest resonance frequencies of the microscope is likely associated with the pendulum modes of the two arms (molybdenum) attaching the sample cubby to the x-motor (shown in Fig. 3.4e). By exciting the the x-motor piezo stacks with a drive voltage of swept frequency and measuring the response in the current, we determine the strongest resonance of the x-motor and sample cubby assembly to be at ∼900 Hz at room temperature, with an additional weaker response at ∼700 Hz. The combination of the two-sample holder cubby, the reliability of the motion of the motors, and the repeatability and precision of the capacitive position sensors provide a critical function when studying samples which cleave poorly. Approaches onto bad areas of the cleavable sample which change the tip do not terminate the experiment, but rather can be simply corrected by field emitting and checking the tip on a simple metal sample in the other slot. After field emission, using the previous position register, we can return to the same spot (macroscopically) on the cleavable sample to continue looking for an acceptable area. Most importantly, even when an acceptable area is found, the microscope can be used to look for an even better area, or for areas with rare surface terminations, with the knowledge that the sensors and motors are reliable enough to allow the microscope to return to the original area should another good area not be found. The ability to systematically search for the best area, or a very rare surface, on a cleavable sample greatly reduces both the number of samples and the time it takes to do an experiment when compared to being limited to examining representative areas, as is the case in most STM instruments. 47
1.8 1.6 1.4 dI/dV (au)
1.2 1 0.8 0.6 0.4 0.2 0 −0.6
∆ = 176 uV T = 240 mK −0.4
−0.2
0 pm
0 Bias (mV)
0.2
3 pm
0.4
0.6
Figure 3.5: This plot shows the differential tunneling conductance of a superconducting Al(100) sample (Tc = 1.16 K) measured with a normal PtIr tip, using a lockin amplifier with an ac modulation of 5 µV at 865 Hz, along with a fit to the thermally broadened BCS density of states with Telectron = 240 mK (green). The spectrum was acquired at a setpoint of 75 pA at -0.6 mV. (Inset) Unfiltered topographic image at base temperature over 30 ˚ A at a setpoint current of 1 nA and bias of -1 mV of the same Al sample showing ∼2 pm amplitude atomic modulations.
3.5
STM Performance
As we have already described, the instrument provides a similar level of functionality to that present in higher temperature STMs. In this last section, we show that it also provides a comparable level of performance, but at significantly lower temperatures, and in high magnetic fields. All data presented are measured using a Nanonis SPM controller and a Femto LCA current preamplifier with 1 kHz bandwidth and 5 · 109 V/A gain. In Figure 3.5, we show data taken on a Al(100) surface prepared in situ using ion sputtering and annealing, and measured at base temperature and zero magnetic field. The topograph in the inset shows well-resolved atoms, even on a challenging material where the atomic corrugation is very small (< 5 pm). Moreover, the differential conductance measured on Al provides a direct measure of 48
a
b
0T
123 nS
5.7 T
135 nS
Figure 3.6: This figure shows two spatial maps of the tunneling conductance, each recorded at a bias of +2 mV, over a field of view of 67 nm square on the heavy fermion superconductor CeCoIn5 [7]. The maps were taken at base temperature on the same area at (a) zero and (b) 5.7 T applied field using an ac modulation of 66 µV at 1.11 kHz. the electron temperature of samples placed in our microscope. This temperature can exceed the measured lattice temperature (20 mK) due to heating from unthermalized electromagnetic radiation transmitted from room temperature to the sample along the electrical line delivering the sample bias [63, 64]. Fitting our tunneling density of states on Al to the thermally broadened density of states for a Bardeen-CooperSchrieffer superconductor ρBCS (E) ∝
√
|E| , E 2 −∆2
where ∆ is the size of the gap, we find
the size of the gap to be ∆ = 176 µV , in agreement with that expected for Al(100) tunnel junctions [65], and the electron temperature of our instrument to be 240 mK. Ideas for reducing this electron temperature are discussed in the conclusion. More importantly, the overall level of noise is low enough to enable us to measure one of the most demanding, but most powerful, kinds of data typically taken using STM: spectral maps, in which the STM is used to visualize spatial patterns in the density of states at a fixed energy with sub-Angstrom spatial resolution. In Figure 3.6, we show two such maps taken at base temperature on the heavy fermion superconductor CeCoIn5 [7], a system in which superconductivity develops below 2.3 K 49
and can be extinguished upon application of a 5 T field at low temperature. The maps are taken at an energy outside the superconducting gap (measured to be 0.5 mV), and demonstrate that the standing waves created by the interference of scattered quasiparticles, which bear the fingerprint of the underlying band structure, are not altered by the application of a field large enough to suppress superconductivity in this material. These maps also indirectly demonstrate the stability of the microscope, which, over the course of several weeks, was able to take two dozen spectral maps at different energies, both at zero field and in applied field, on the same area of the sample. The data in Figures 3.5 and 3.6 are qualitative proof of the level of performance of this instrument. A more quantitative figure of merit for any STM, one which can be used to compare the relative levels of performance of different instruments, or different modes of operation, is the integral of the spectral density of the current over low frequencies. The limiting noise in STM measurements is invariably low-frequency noise because it is time-consuming to average out and shows up in both topographic and spectroscopic measurements. In Figure 3.7, we compare the spectral noise density of the tunneling current for this instrument and another STM in the lab which uses springs and magnetic damping right at the microscope head itself to isolate vibrations. Notably as shown in Figure 3.7a, this instrument has a comparable amount of low frequency noise (in orange; integral over 125 Hz is 0.49 pA2 ) to the more conventional design (in green; 0.28 pA2 over 125 Hz) when the system runs with the 1K pot in single-shot mode and the dilution refrigerator off. The amount of low-frequency noise when also running the dilution refrigerator continuously, while considerably larger (in blue; 2.5 pA2 over 125 Hz), is directly comparable to published measurements from the UHV dilution refrigerator STM of Ref. [9] (∼ 3 pA2 over 100 Hz, using Fig. 19). However, in our case, the system must be run with the 1K pot in single-shot mode, which means it must be refilled every 8 hours, in order to achieve this level of noise. 50
a
Current SD (A/√Hz)
10
10
10
10
Base Temperature 1K Mode (T ~ 2 K) Tip Retracted Compare: STM after Ref. 17
−12
−13
−14
−15
0
25
50 75 Frequency (Hz)
100
125
1000
1250
b
Current SD (A/√Hz)
10
10
10
10
Base Temperature 1K Mode (T ~ 2 K) Tip Retracted Compare: STM after Ref. 17
−12
−13
−14
−15
0
250
500 750 Frequency (Hz)
Figure 3.7: These are plots of the open feedback noise on the tunneling current over two frequency ranges (125 Hz in (a), 1250 Hz in (b)) with the 1K pot running in single shot-mode (orange) and the dilution refrigerator circulating with the 1K pot in single-shot mode (blue). Also shown is the open feedback noise on an STM based on the design of Ref. [66] (green). The data were taken with a DC tunneling current of 100 pA and a bias of -200 mV on a clean Cu (111) surface. The black curve is the noise on the current with the tip outside of tunneling range, and serves as a measure of the bare amplifier noise due to stray capacitance.
51
While the spectral noise density contains many seemingly deleterious resonances at higher frequencies, as shown in Figure 3.7b, these have a minimal impact on our measurements in practice. Most occur at frequencies far above the bandwidth of the feedback loop, and hence don’t appear in topographs or as a set-point error in spectroscopic measurements. Moreover, the lockin oscillator frequency can still be set to frequencies where the noise spectrum is no worse than the amplifier background (near ∼1100 Hz, for example). Evidence suggests that much of the relevant noise originates in the 1K pot itself. The level of noise with both the dilution refrigerator and 1K pot running single-shot is comparable to that when running only the 1K pot single-shot with the dilution refrigerator off. Conversely, the level of noise when running the 1K pot single-shot and dilution refrigerator continuously is comparable to that when running the 1K pot continuously and leaving the dilution refrigerator off.
3.6
Conclusion
The recent development of this and other dilution refrigerator STMs opens the door to studying exotic electronic phases and quantum phenomena which only occur at milli-Kelvin temperatures with a spectroscopic tool and at the atomic scale. We have described the construction of an instrument which extends both the functionality and level of performance present in higher temperature STMs down to dilution refrigerator temperatures. As the current measured electron temperature of 240 mK is an order of magnitude larger than the measured lattice temperature of 20 mK, further improvements to thermalizing the tip and sample electron temperature can be made. For example, increasing the cooling power delivered to the sample cubby itself (currently thermally anchored through only a thin strip of silver foil 0.4 mm2 in
52
cross-section to facilitate sample motion)1 and adding more stages of low temperature RF filters [63, 64] on all of the electrical lines are planned. Only the tip and bias lines are currently filtered at the mixing chamber; all other lines to the microscope are filtered externally at room temperature, potentially causing heating via crossradiation at the microscope. Moreover, for ease of access, the system has never been operated with the RF enclosure fully closed, which may be necessary for the lowest electron temperatures. Finally, a reduction of the noise likely created inside the 1K pot would reduce the time it takes to acquire the high resolution data shown here, which is limited to being taken in 8 hour intervals due to the single-shot lifetime of the 1K pot. Ideally, a set of two dozen spectral maps could be taken in a handful of days, as is common in more conventional instruments, instead of a couple of weeks.
1
Recently, the foil strip was upgraded from 0.300”x0.002” 4N silver to 0.300”x0.004” 5N gold, the sample arms were changed to non-superconducting BeCu, and the 1K rotary door was upgraded to overlap (shield) more when closed. These changes resulted in a minimal lowering the electron temperature from 250 mK to 240 mK, pointing to another factor as the dominant source of the elevated temperature. Inadequate RF filtering and the poor thermal conductivity of the 0.001” thick Kapton strip that electrically isolates the sample cubby thermal anchor are now thought to be the remaining culprits.
53
Chapter 4 Visualizing d-Wave Heavy Fermion Superconductivity in CeCoIn5 This chapter is based upon the publication: Zhou B. B., Misra S. et al., “Visualizing nodal heavy fermion superconductivity in CeCoIn5 .” Nature Phys. 9, 474 (2013) [7].
Understanding the origin of superconductivity in strongly correlated electron systems continues to be at the forefront of unsolved problems in all of physics [67]. Among the heavy f -electron systems, CeCoIn5 is one of the most fascinating, as it shares many of the characteristics of correlated d-electron high-Tc cuprate and pnictide superconductors [68, 69, 70], including the competition between antiferromagnetism and superconductivity [71]. While there has been evidence for unconventional pairing in this compound [29, 30, 72, 73, 74, 75], high-resolution spectroscopic measurements of the superconducting state have been lacking. Previously, we have used high-resolution scanning tunneling microscopy (STM) techniques to visualize the emergence of heavy-fermion excitations in CeCoIn5 and demonstrate the composite nature of these excitations well above Tc [21]. Here we extend these techniques 54
to much lower temperatures to investigate how superconductivity develops within a strongly correlated band of composite excitations. We find the spectrum of heavy excitations to be strongly modified just prior to the onset of superconductivity by a suppression of the spectral weight near the Fermi energy (EF ), reminiscent of the pseudogap state [76, 77] in the cuprates. By measuring the response of superconductivity to various perturbations, through both quasiparticle interference and local pair-breaking experiments, we demonstrate the nodal d-wave character of superconducting pairing in CeCoIn5 .
4.1
Introduction
CeCoIn5 undergoes a superconducting transition at 2.3 K. Despite evidence of unconventional pairing, consensus on the mechanism of pairing and direct experimental verification of the order parameter symmetry are still lacking [29, 30, 72, 73, 75]. Moreover, experiments have suggested that superconductivity in this compound emerges from a state of unconventional quasiparticle excitations with a pseudogap phase similar to that found in underdoped high-Tc cuprates [78, 79, 80]. Previously, we demonstrated that STM spectroscopic techniques can be used to directly visualize the emergence of heavy fermion excitations in CeCoIn5 and their quantum critical nature [21]. Through these measurements, we also demonstrated the composite nature of heavy quasiparticles and showed their band formation as the f -electrons hybridize with the spd-electrons starting at 70 K, well above Tc [21]. This previous breakthrough, together with our recent development of a high-resolution milli-Kelvin STM, offers a unique opportunity to measure how superconductivity emerges in a heavy electron system.
55
a
Surface A
b
Surface B
c
a
b
G(V)/GS
a b
0.8 0.6 0.4 T = 245 mK 0.2
e
0.9
∆SC
0.7 0.6
−30 −20 −10
0
10
Energy (mV)
20
30
0.5
1
1.5
∆SC
1
0.8
Surface B
15
1.2
2.5 K Gs = 1 nS 245 mK Gs = 5 nS
0
f
1.4
0.8 H = 0 T
−1 −0.5
Energy (mV)
G(V) (nS)
G(V)/GS
1
∆PG
1.6
G(V)/G(V = -30 mV)
Surface A
Surface B, 0 T, Gs = 125 nS Surface B, 5.7 T, Gs = 85 nS Surface A, 0 T, Gs = 63 nS
Surface B 1.8
1.1 ∆HG
0 −1.5
5 nm
5 nm
d
∆SC
1
∆PG
13
∆SC
11 5.7 T
H=0T 7.2 K
1.7 K
5.3 K
245 mK
0.6 −30 −20 −10
0
10
Energy (mV)
9 −10
0T
−5
T = 245 mK
0
5
10
Energy (mV) 20
30
Figure 4.1: Topographic image measured on surface A (a) and on surface B (b) of CeCoIn5 at 245 mK. Insets in (a) and (b) zoom in on 12 by 12 nm2 regions on their respective surfaces. The arrows in the figure indicate the in-plane crystallographic a and b directions. (d,e) Corresponding conductance spectra G(V ), proportional to the local electronic density of states on surface A and B carried out at temperatures above and below Tc , showing the evolution of the different energy scales (∆HG : hybridization gap; ∆P G : pseudogap; ∆SC : superconducting gap) with temperature. Spectra are offset for clarity in (e). (c,f) Blow up of the superconducting gap energy scale showing the destruction of the superconducting gap in a magnetic field of H = 5.7 T > Hc2 while the pseudogap feature is preserved. The spectra G(V ) in (c) and (d) are normalized by their corresponding junction impedances GS .
4.2
Superconductivity on the Two Surfaces of CeCoIn5
Figure 4.1 shows STM topographs of the two commonly observed atomically ordered surfaces of CeCoIn5 produced after the cleaving of single crystals in situ in the ultra-high vacuum environment of our milli-Kelvin STM. We have previously 56
shown through experiments and theoretical modeling that different surface terminations change the coupling between the tunneling electrons and the composite heavy fermion excitations in this compound [21]. Tunneling into such composite states can be influenced not only by the coupling of the tip to spd- or f -like component of such states but also by the interference between these two tunneling processes. On surface A, tunneling measurements are more sensitive to the lighter component of the composite band structure, and accordingly, the spectra show evidence for a hybridization gap centered at +9 mV, as shown in Fig. 4.1 d). At temperatures below Tc , this hybridization gap is modified by the onset of an energy gap associated with superconductivity (Fig. 4.1 c,d), as further confirmed by its suppression with the application of a magnetic field larger than the bulk upper critical field (Hc2 = 5.0 T perpendicular to the basal plane of this tetragonal system) of CeCoIn5 (see Fig. 4.2). Instead of focusing on measurements of surface A, where the tunneling is dominated by the lighter part of the composite band, we turn to measurements of surface B. On this surface tunneling directly probes narrow bands of heavy excitations which result in a peak in the density of states near EF (Fig. 4.1 e). Lowering the temperature from 7.2 K to 5.3 K, above Tc , we find that this peak is modified by the onset of a pseudogap-like feature at a smaller energy scale. Further cooling shows the onset of a distinct superconducting gap below Tc inside the pseudogap. Measurements in a magnetic field corroborate our finding that the lowest energy scale on surface B (∼ ±500 µV, as shown in Fig. 4.1 c) is indeed associated with pairing, as it disappears above Hc2 , while the intermediate energy scale pseudogap remains present at low temperature in the absence of superconductivity at high magnetic field (Fig. 4.1 f). This behavior is reminiscent of the pseudogap found in underdoped cuprates, where the superconducting gap opens inside an energy scale describing strong correlations that onset above Tc . However, unlike cuprates, here we clearly distinguish between the two energy scales by performing high-resolution spectroscopy in a mag57
a
b
Figure 4.2: a) The conductance spectra show a sudden shift in their zero bias conductances at Hc2 , resembling a 1st order phase transition from the superconducting into the normal state, consistent with bulk transport measurements in CeCoIn5 . b) The zero bias conductance G(V = 0) as a function of magnetic field across Hc2 netic field large enough to fully suppress superconductivity. Detailed measurements of changes in the spectra with the magnetic field also confirm that the transition out of the superconducting state at Hc2 is first order (Fig. 4.2), showing that our measurements are consistent with the bulk phase diagram of CeCoIn5 . The spectroscopic measurements suggest that electronic or magnetic correlations alter the spectrum of heavy excitations by producing a pseudogap within which pairing takes place. These measurements also show the shapes of the spectra at the lowest temperature to be most consistent with a d-wave superconducting gap, as they have a nearly linear density of states near zero energy (Fig. 4.1 c). However, measurements on all surfaces and on several samples reveal that this d-wave gap (with a magnitude of 535±35 µV, consistent with that extracted from point contact data [81, 82] is filled (40%) with low energy excitations - a feature that cannot be explained by simple thermal broadening (determined to be 245 mK from measurements on a single-crystal Al sample). The complex multiband structure of CeCoIn5 could involve different gaps on different Fermi surface sheets, and there is the possibility that some remain ungapped even at temperatures well below Tc [83]. Another contribution to the in-gap density of states could come from surface impurities, since even non-magnetic impurities per58
turb a nodal superconductor, as we demonstrate below. Before we address the nature of the in-gap excitations, we first demonstrate in more detail the connection between pairing and the heavy fermionic states of CeCoIn5 .
4.3
Quasiparticle Interference in Normal and Superconducting States of CeCoIn5
Energy-resolved spectroscopic mapping with the STM can be used to measure the interference of quasiparticles (QPI) in order to examine the heavy Fermi surface. As shown in Fig. 4.3 a)-d), features in the discrete Fourier transform (DFT) of these maps show wavevectors related to the elastic momentum transfer Q(E), connecting the initial and the final momentum states on the contours of constant energy. Previous theoretical calculations, quantum oscillation, and angle resolved photoemission spectroscopy measurements have shown CeCoIn5 to have a complex three-dimensional band structure, with the α and β bands being the most relevant near EF (Fig. 4.3 e) [84, 85, 86]. Our previous QPI measurements on surface A show features that are most consistent with 2kF scattering originating from the α band. The QPI measurements presented here on surface B (outside of the superconducting energy scale) display scattering wavevectors originating from a larger Fermi surface volume and are more consistent with scattering involving the β band. Since QPI does not probe the Fermi surface directly, inferring a unique Fermi surface in a three-dimensional, multi-band material without making large number of assumptions is not possible. Nevertheless, the results of QPI measurements (Fig. 4.3 a-d) together with spectroscopic measurements (Fig. 4.3 e) demonstrate that the superconducting instability occurs within a correlated heavy quasiparticle band of CeCoIn5 with a large density of states at the Fermi energy (for additional details see Appendix C).
59
Q2
b
10 nm
a
+8 nS
4
Q1
2
1.5 mV c
b
3.0 mV
PSD
0
d
Q2 Q3
α β Q3 Q2
Q1
0.1 0.2 0.3
Q1,2,3 (rlu)
Q1
e
Q3
f
g
h
i
j
Normal H > Hc
Superconducting H=0
-8 nS
Energy (mV)
Q3
a
k
l -255 µV
m -85 µV
n 0 µV
o +85 µV
+255 µV
Figure 4.3: Real space conductance map (a) and its DFT (b) at a bias of 1.5 mV measured at T = 245 mK on surface B. Colorbar in (a) denotes deviation from the mean. Q1 , Q2 , Q3 correspond to the different quasiparticle scattering vectors. (c) DFT at V = 3 mV. Axes in (c) denote the Bragg orientation for all DFTs and for the schematic (e). (d) Energy-momentum structure of Q1 , Q2 , Q3 showing rapid dispersions reflective of mass enhancements m* = 34 m0 , 29 m0 , 23 m0 respectively. Error bars are derived from the width of the peaks in the DFTs. (e) Schematic of the band structure in the first Brillouin zone derived from Refs. [84, 85, 86] showing the α (magenta), β (blue) and small (orange) Fermi surfaces in the kz = 0 (solid) and kz = π (dashed) planes. The measured Q1 , Q2 & Q3 QPI scattering vectors are drawn to scale for comparison. DFTs for selected energies in the superconducting (f-j) and normal (k-o) states. The Q-space range of the DFTs in (b,c,f-o) is ±0.5 rlu, where 1 rlu = 2π/a0 = 2π/4.6 ˚ A.
60
We focus our discussion next on the momentum structure of the superconducting gap, first by examining the conductance maps within this smaller energy window on the same area of the sample (with the same tip) in the normal (H > Hc2 ) and superconducting (H = 0) states of CeCoIn5 . As the data in Fig. 4.3 f)-o) demonstrate, we observe clear differences between the DFT maps in the superconducting (H = 0) and normal states (H = 5.7 T). Typically, quasiparticle interference at low energies in a superconductor is associated with the scattering of Bogoliubov-de Gennes (BdG) excitations and is often analyzed to obtain information about the momentum structure of the superconducting gap [45, 87, 88]. In particular, contrasting the zero-energy DFTs in the superconducting (Fig. 4.3 h) and normal (Fig. 4.3 m) states, we see an enhancement of quasiparticle interference at wavevector Q3 (see also Fig. 4.4), suggestive of nodal BdG quasiparticles in a dx2 −y2 superconductor. However, if such features were only due to BdG-QPI, then they should display a particle-hole symmetric dispersion in their energy-momentum structure away from the nodes, as seen for example in similar measurements of high-Tc cuprates [45]. The absence of such particle-hole symmetry in our data (Fig. 4.3 f-j) together with the large zero-bias density of states (40%, see Fig. 4.1 c) suggests that such QPI measurements are complicated by an ungapped portion of the Fermi surface or by in-gap impurity-induced states, which are expected to have a particle-hole asymmetric structure (see measurements & discussion below). These complications together with complex three-dimensional nature of the Fermi surface of this compound makes extraction of the gap function from such QPI measurements unreliable (see Appendix C).
61
Figure 4.4: While the overall differential conductance at zero bias is reduced due to the superconducting gap, the QPI strength at Q3 (labeled in Fig. 4.3), in contrast, is enhanced. Subtraction of the peak power spectral density of the Q3 feature in the normal state from the superconducting state shows enhancement centered at zero energy, consistent with the confinement of the Fermi surface around the (π, π) nodes by a dx2 −y2 superconducting gap.
4.4
Response of Nodal Superconductivity to Potential Scattering
In contrast, using the power of STM to probe the real space structure of electronic states, it is still possible to find direct spatial signatures of the nodal character of superconductivity in CeCoIn5 that do not require multi-parameter modeling or ad hoc assumptions to interpret. The first such signature can be found by examining the response of low-energy excitations to extended potential defects such as atomic step edges. Spectroscopic mapping with the STM upon approaching such steps shows direct evidence for the suppression of superconductivity in their immediate vicinity (Fig. 4.5 a-b). This suppression is consistent with the expected response of a nodal superconductor to non-magnetic scattering (Fig. 4.5 c), analogous to similar observations in the cuprates [89], and in marked contrast with our step-edge measurements of the conventional s-wave superconductor Al (see Fig. 4.6). The data in Fig. 4.5 d) provide a direct measure of the Bardeen-Cooper-Schriefer (BCS) coherence length 62
ξBCS = 56 ± 10 ˚ A, in agreement with ξBCS ∼ ~vF /π∆ ∼ 60 ˚ A using the gap observed in Fig. 4.1 (0.5 meV) and the Fermi velocity extracted from Fig. 4.3 (1.5 · 106 cm/s)
5 nm
c
d
+
_
_
+
G0(r) - G0(r = ∞) (nS)
Lower Step
b
b a
a
G(V) - G(V, r = 153 Å) (nS)
[90]. 0Å 25 Å 51 Å 76 Å 102 Å
10 5 0 -5 -2
-1
0 1 Energy (mV)
2
y = a*exp(-x/ξ) ξBCS = 56 � 10 Å
10
1 0
50 100 Distance (Å)
150
Figure 4.5: Topographic image (V = -100 mV, I = 100 pA) of surface A showing a single unit-cell step-edge oriented at 45◦ to the atomic lattice. The arrows in the figure indicate the in-plane crystallographic a and b directions (b) Evolution of the spectra near the step-edge: G(V ) subtracted by the spectrum far away from the step-edge G(V, r = 153˚ A). The locations of the spectra in (b) are plotted on (a). (c) Schematic representation of nodal superconducting quasiparticles scattering off a step-edge. (d) Zero-bias conductance G0 (r) subtracted by the extrapolated G0 (r = ∞) as a function of distance from the step edge. Line represents an exponential fit to the data, where error bars denote the standard deviation on the averaged spectra. ξBCS denotes the characteristic decay length obtained from the fit in (d), which is a measure of the BCS coherence length. .
63
a
b
Figure 4.6: In contrast, the gaps measured on Al(100) single crystal show no sensitivity to the proximity to the step edge, consistent with the s-wave nature of the superconducting gap, which is robust against potential scattering by Anderson’s theorem [91].
4.5
Vortex Anisotropy
Application of a magnetic field can also be used to probe the local suppression of heavy-fermion superconductivity in CeCoIn5 due to the presence of vortices and the Abrikosov lattice. As shown in Fig. 4.7 a-b, STM conductance maps can be used to directly visualize the vortex lattice in this compound, which can have different structures depending on the magnetic field. Such structural changes of the vortex lattice (transition between rhombic and square lattices) have been previously studied in neutron scattering experiments [92] and various theoretical models [93]. Complementing these efforts, the STM can be used to probe the electronic states within the vortex core directly, as shown in Fig. 4.7 d, to demonstrate the presence of a zeroenergy vortex bound state. Analysis of this core state demonstrates the anisotropic decay of the vortex bound state (Fig. 4.7 c,e), the angular average (Fig. 4.7 e) of which determines the Ginzburg-Landau coherence length scale ξGL = 48 ± 4 ˚ A, consistent with an independent estimate from |dHc2 /dT |(T =Tc ) [94].1 While observation 1
One can relate both quantities mentioned in this sentence to the estimated orbital critical field 2 Hc2,orbital . We estimate Hc2,orbital = Φ0 /(2πξGL ) ∼ 14.5 T from our measurement. Accordingly, for
64
of such anisotropy is consistent with the nodal character of pairing, an understanding of the role of the underlying Fermi surface symmetry and vortex-vortex interactions is required to model the STM data in more detail.2 85 nS
135 nS
a
d
40 - 65 nS
θ = 45º to b-axis
b 200 nm x 120 nm
b
84
100
(r, θ)
a
e G0(θ)
c
0]
10 b[
0
75
80 76 [100] [010] 0 90 180
270
y = a*exp(-x/ξ) ξGL = 48 � 4 Å
80
θ (º)
50
25
r ‖ [110]
40
(Å)
20
G0(r) - G0(r = ∞) (nS)
a
a
60
b
10 nm
−2
−1
0
1
Energy (mV)
2
80
0
20
40
60
r (Å)
Figure 4.7: Zero-bias conductance maps both taken at H = 1 T (separate field dials) and at T = 245 mK show the vortex lattice structure expected below (a) and above (b) the transition seen at this field by neutron scattering in Ref. [92]. The arrows in the figure indicate the in-plane crystallographic a and b directions (c) Close-up zero-bias map of the vortex lattice on surface B showing an anisotropic square vortex core (H = 1.5 T). (d) Line-cut of spectra starting from the center of a vortex and moving radially outward at 45 degrees to the b-axis showing the evolution of the bound state inside the superconducting gap (H = 0.5 T). (e) Radial dependence of the angularly averaged zero-bias conductance G0 for a single vortex core at H = 1 T. Error bars (estimated from the standard deviation in the analyzed map) are smaller than the marker size in (e). Inset shows the angular dependence of the radially averaged conductance showing the four-fold anisotropy of a single vortex with higher conductance extending along the a- and b-directions directions. ξGL denotes the characteristic decay length obtained from the fit in (e), which is a measure of the angularly averaged Ginzburg-Landau coherence length. a BCS superconductor in the clean limit, Hc2,orbital (T = 0) = −0.73|dHc2 /dT |(T =Tc ) ∼ 15 T, using the value reported in Ref. [94]. It is an open question why the BCS formalism still applies rather well for a d-wave superconductor such as CeCoIn5 . 2 Specifically, the extent of the vortex bound state farther along the crystallographic a, b directions is inconsistent with the naive expectation based on purely gap symmetry considerations for a dx2 −y2 superconductor [93], where the state at zero energy is expected to leak out along the diagonal nodes. This demonstrates that the other effects of band structure and vortex-vortex interactions must be more important in CeCoIn5 .
65
4.6
Impurity Bound State: Fingerprint of dx2−y2 Pairing
A more spectacular demonstration of the nodal pairing character in CeCoIn5 can be obtained from examining the spatial structure of in-gap states associated with defects on the surface of cleaved samples. The spatial structure of impurity quasibound states, which are mixtures of electron-like and hole-like states, can be a direct probe of the order parameter symmetry [90, 95]. Figure 4.8 shows an extended defect with a four-fold symmetric structure, which perturbs the low energy excitations of CeCoIn5 by inducing an in-gap state. As shown in Fig. 4.9, this particular impurity attracts the electron part of the BdG quasi-particle, and hence its hole counterpart, linked by the backdrop of Cooper pairing, will revolve around the impurity with the symmetry of the pairing potential. The spatial distribution of the hole (electron) state can be imaged directly with STM by applying a negative (positive) bias. Indeed, probing the spatial structure of these impurity states, we not only find their expected electron-hole asymmetry, but also find that their orientation is consistent with that predicted for a dx2 −y2 superconductor (Fig. 4.8 b-e) [95]. The minima (maxima) in the oscillations for hole-like (electron-like) states identify the nodes of the d-wave order parameter to occur at 45◦ to the atomic axes (Fig. 4.8 h). In fact, these features in the STM conductance maps are identical to those associated with Ni impurities in high-Tc cuprates [90, 96]. However, in contrast to measurements in the cuprates, we are able to determine the spatial structure that such impurities induce on the normal state by suppressing pairing at high magnetic fields. Such measurements allow us to exclude the influences of the normal state band structure, of the impurity shape, or of the tunneling matrix element [90] on the spatial symmetries of the impurity bound state in the superconducting state. Contrasting such measurements for H > Hc2 (in Fig. 4.8 f-g) with measurements on the same impurity for H = 0 (Fig. 4.8 d-e) we 66
directly visualize how nodal superconductivity in CeCoIn5 breaks the symmetry of
0 pm
5 pm
a
4 a0 b
Calculation
1
hole-like
0.51
∆ / ∆max
a
G(±V) / [G(+V)+G(-V)]
the normal electronic states in the vicinity of a single atomic defect.
0.50
0.49
h
electron-like 0
90
180
270
Angle from a axis (degrees) H > Hc2 H=0
0 360
- 195 µV
d
f
c
e
g
+ 195 µV
b
low
high -7 nS
+7 nS
Figure 4.8: (a) Topographic image of an impurity on surface B. (b) Model calculation for the real space structure (roughly 10 Fermi wavelengths across) of the hole-like part of the impurity bound state in a dx2 −y2 superconductor, reproduced from Ref. [95] (Copyright (2000) by the American Physical Society). (c) Electron-like state for the same impurity in (b). (d-g) Local density of states obtained on the same field-of-view as (a) at ±195 µV in the normal (H > Hc2 ) and superconducting (H = 0) states as indicated on the figure. Colorbar in (d-g) denotes deviation from the mean. (h) Radial average of the density of states across the lobes measured in (d,e), normalized to their sum, as a function of angle from the b axis. Data at negative (positive) energy is shown in blue (red) symbols; the lines are guides to the eye. A dx2 −y2 gap is shown in yellow. 67
Norm. Conductance (a.u.)
hole
electron
1.1
1
0.9 −1
−0.5 0 0.5 Energy (mV)
1
Figure 4.9: The spectrum on top of the impurity in CeCoIn5 (cyan dot in Fig. 4.8 a), normalized by the spectrum far away from the impurity, shows that the impurity attracts the electron component of the Cooper pair as its spectrum shows an enhancement at positive energies.
4.7
Outlook
The appearance of a pseudogap and the direct evidence for dx2 −y2 superconductivity reported here together with previous observations of the competition between antiferromagnetism and superconductivity closely ties the phenomenology of the Ce-115 system to that of the high-temperature cuprate superconductors. An important next step in extending this phenomenology would be to explore how the competition between anti-ferromagnetism and superconductivity manifests itself on the atomic scale in STM measurements. Similarly, extending our studies of the electronic structure in magnetic vortices could be used to examine the competition between different types of ordering in the mixed state, and the possible development of the Fulde-FerrellLarkin-Ovchinnikov state in this Pauli-limited superconductor [92, 97, 98]. Note added in proof of dissertation: The ‘pseudogap’ feature observed in the spectra of CeCoIn5 could alternatively be a manifestation of the Fermi surface singularity seen in the Gutzwiller projection of a strongly-interacting metal, as pointed out by Phil Anderson in Nature Phys. 2, 626-630 (2006). Hence, this feature may not necessarily require proximity to competing order at a quantum critical point. 68
Chapter 5 The Three Dimensional Dirac Semimetal Cd3As2 This chapter is based upon the publication: Jeon S., Zhou B. B. et al., “Landau quantization and quasiparticle interference in the three-dimensional Dirac semimetal Cd3 As2 .” Accepted for publication in Nature Mater. http://dx.doi.org/10.1038/nmat4023 (2014) [39].
Condensed matter systems provide a rich setting to realize Dirac [99] and Majorana [100] fermionic excitations and the possibility to manipulate them in materials for potential applications [101, 102]. Recently, it has been proposed that Weyl fermions, which are chiral, massless particles, can emerge in certain bulk materials [34, 103] or in topological insulator multilayers [104] and can produce unusual transport properties, such as charge pumping driven by a chiral anomaly [35]. A pair of Weyl fermions protected by crystalline symmetry [33], effectively forming a massless Dirac fermion, has been predicted to appear as low energy excitations in a number of candidate materials termed three-dimensional (3D) Dirac semimetals [33, 40, 41]. Here we report scanning tunneling microscopy (STM) measurements at 69
sub-Kelvin temperatures and high magnetic fields on one promising host material, the II-V semiconductor Cd3 As2 . Our study provides the first atomic scale probe of Cd3 As2 , showing that defects mostly influence the valence band, consistent with the observation of ultra-high mobility carriers in the conduction band. By combining Landau level spectroscopy and quasiparticle interference (QPI), we distinguish a large spin-splitting of the conduction band in a magnetic field and its extended Dirac-like dispersion above the expected regime. A model band structure consistent with our experimental findings suggests that for a specific orientation of the applied magnetic field, Weyl fermions are the low-energy excitations in Cd3 As2 .
5.1
Introduction
One starting point for accessing the novel phenomena of Weyl fermions is the identification of bulk materials with 3D Dirac points near which the electronic dispersion is linear in all three dimensions [105] in analogy to 2D Dirac points observed in graphene [99] or topological insulators [31]. With time reversal and inversion symmetries preserved, 3D Dirac points can be formed at the crossing of two doubly degenerate bands and constitute two overlapping Weyl points. However, 3D Dirac points are generally not robust to gapping unless they occur along special high symmetry directions in the Brillouin zone, where the band crossing is protected by crystalline point group symmetry [33, 40, 41]. In these 3D Dirac semimetals, individual Weyl nodes can be isolated only by breaking either time reversal or inversion symmetry. Since Weyl nodes are topological objects of definite helicity, acting as either a source or sink of the Berry curvature, they are robust against external perturbation and are predicted to harbor exotic effects, such as Fermi arc surface states [34] and chiral, anomalous magneto-transport [35, 106]. These unusual transport phenomena of Weyl fermions have been proposed as the basis for novel electronic applications [38, 107]. 70
Several candidate materials, including Na3 Bi and Cd3 As2 , were recently predicted [40, 41] to exhibit a bulk 3D Dirac semimetal phase with two Dirac points along the kz axis, stabilized by discrete rotational symmetry. While photoemission measurements [108, 109, 110, 111] indeed observed conical dispersions away from certain points in the Brillouin zone of these materials, high energy resolution, atomically-resolved spectroscopic measurements are needed to isolate the physics near the Dirac point and clarify the effect of material inhomogeneity on the low-energy Dirac behavior. Low-temperature scanning tunneling microscopy experiments are therefore ideally suited to address these crucial details. Previously, Cd3 As2 has drawn attention for device applications due to its extremely high room temperature electron mobility [112] (15,000 cm2 /V s), small optical band gap [112], and magnetoresistive properties [113]. The recent recognition that inverted band ordering driven by spin-orbit coupling can foster nontrivial band topology renewed interest in Cd3 As2 , which is the only II3 V2 semiconductor believed to have inverted bands. Updated ab initio calculations predict 3D Dirac points formed by shallow band inversion between the conduction s-states, of mainly Cd-5s character, and the heavy hole p-states, of mainly As-4p character [41, 114]. However, the large unit cell of Cd3 As2 with up to 160 atoms due to Cd ordering in a distorted anti-fluorite structure [114] present complications to first-principles calculations, which must be corroborated by careful experimental measurement of the band structure.
5.2
Topographic and Spectroscopic Characterization at Zero Field
To probe the unique electronic structure of Cd3 As2 , we perform measurements in a home-built low-temperature STM [10] capable of operating in magnetic fields up to 14 T. Single crystal Cd3 As2 samples are cleaved in ultra-high vacuum and cooled to an 71
electron temperature of 400 mK, where all spectroscopic measurements described here are performed. Figure 5.1 a) and its inset show an atomically-ordered topography of a cleaved surface and its associated discrete Fourier transform (DFT). The pseudohexagonal Bragg peaks, circled in red, reveal a nearest-neighbor atomic spacing of 4.4±0.15 ˚ A. Their magnitude and orientation precisely match the As-As or Cd-Cd spacing in the (112) plane of this structure [114], schematically illustrated in Fig. 5.1 b), and identify this facet as a natural cleavage plane for Cd3 As2 . Because we image atoms at ∼96% of the sites in the pseudo-hexagonal lattice, we further attribute the cleaved surface to an As layer, since any Cd layer would contain ∼25% empty sites in this projection. We present in Fig. 5.1 c) the tunneling differential conductance (proportional to the local DOS) measured at B = 0 T along a line spanning 30 nm. Photoemission measurements [108, 109] locate the Dirac point (EDirac ) for naturally grown Cd3 As2 at -200±20 mV, corresponding to a carrier concentration ne ∼ 2 · 1018 cm−3 . In agreement, the STM conductance spectra show a depression near this energy, and the measured DOS rises as (E − EDirac )2 away from it as expected for 3D Dirac points [115]. The conductance near the Dirac point is nonzero and smooth, representative of a semi-metallic band crossing rather than a band gap. While the presence of surface states can mask a bulk gap [48], we rule out this possibility by performing QPI measurements, shown below, that do not resolve a strong surface state signal near EDirac . The absence of a gap, particularly at the low temperature of our measurement, is consistent with the proposed theoretical description shown in the inset of Fig. 5.1 c) and d), which illustrate a shallow inversion between the valence and conduction bands. Additionally, the zero-field spectra in Fig. 5.1 c) display significant spatial fluctuation for energies below EDirac , while in contrast, they are highly homogeneous for energies above EDirac . Since the carrier concentration in as-grown Cd3 As2 is attributed to As vacancies [116], these lattice defects would be expected to primarily 72
a
b
c
[001] direction
4
dI/dV (nS)
d
e
Dirac Points
3
Dirac Point
2 1 0
−300
−200
−100
0
Energy (meV)
100
200
Figure 5.1: Crystal and band structures of Cd3 As2 (112) cleaved crystal. a) Atomically-ordered topographic image (I=50 pA, V=-250 mV) of the Cd3 As2 (112) surface. Inset shows its 2D Fourier transform. Red circles are associated with Bragg peaks and blue circles with reconstruction peaks. b) Schematic of the Cd3 As2 unit cell along the (112) plane (red). Cd atoms and As atoms both make a pseudo-hexagonal lattice. c) Differential conductance spectra (I=300pA, V=250mV) taken at 90 spatial positions over a line spanning 30 nm. The blue curves show the individual spectra and the red curve is the spatial average. Spatial variation in the local density of states is especially pronounced below the Dirac point. The inset shows the schematic band dispersion along the [001] direction passing the Γ point. d) Schematic band structure + − of Cd3 As2 based on ab initio calculations. Two 3D Dirac points marked as kD , kD are located along the [001] direction and are evenly separated from the Γ point. The k⊥ direction refers to any axis perpendicular to the kz direction. e) Schematic of the Fermi surfaces above (red) and below (blue) the Lifshitz transition. The overlaid solid curves represent the extremal cross-sections parallel to the (112) plane, showing two pockets merging into a single ellipsoidal contour.
73
Figure S2 a
0
5Å
b
0
dI/dv(r)/mean
3
c
0
dI/dv(r)/mean
3
100 Å
E = -500 mV
E = 0 mV
(μ = 154 pS, σ = 52 pS, σ/μ = 0.34)
(μ = 49 pS, σ = 10 pS, σ/μ = 0.21)
Figure 5.2: Stronger conductance variation is seen at energies in the valence band b), than at energies in the conduction band c). Surface defects in the As plane imaged as depressions in the topography a) localize strong enhancements in conductance in the hole-like band, while they negligibly impact the conductance at the Fermi level. impact the valence band rather than the conduction band. In Fig. 5.2, we show that a common, clustered defect in the As plane (visible as the dark depressions) produces strong fluctuations in the conductance of the valence band, but is virtually invisible at the Fermi level. This microscopic information may explain the broad valence band seen in photoemission measurements [108, 109] and the high mobility at the Fermi level [112], and suggests routes for further materials optimization.
5.3
Landau Level Spectroscopy
Landau level spectroscopy with STM has previously been applied to extract precise band structure information for graphene [117, 11], semiconductor 2D electron gases [118], and topological insulator surface states [119, 120]. Here in distinction, we extend this technique to quantify the bulk 3D dispersion of Cd3 As2 by applying a magnetic field perpendicular to the cleaved (112) surface of the sample. The 3D band structure is quantized by the magnetic field into effectively 1D Landau bands that disperse along the momentum k3 parallel to the field. The projected bulk DOS 74
measured by STM is an integration over all k3 and accordingly displays peaks at the minimum or maximum energies of these Landau bands, which contribute inverse square root divergences to the DOS. Semi-classically, these extrema describe Landau orbits along the constant energy contours of the band structure with extremal crosssectional area perpendicular to the magnetic field. In Fig. 5.1 e), we illustrate the extremal contours parallel to the (112) plane in Cd3 As2 for energies above and below the Lifshitz transition, demonstrating the merging of two Dirac pockets into a single ellipsoidal contour. Figure 5.3 a) illustrates the Landau level fan diagram for Cd3 As2 assembled from spectra measured from 0 to 14 T at a single fixed location on the sample surface. Four aspects are immediately striking. First, the Landau levels emanate from a point slightly below -200 mV, revealing the presence of a band extremum in the vicinity of the Dirac point determined by photoemission. This suggests that the band inversion is small, consistent with ab initio predictions. Second, all prominent Landau levels are electron-like, dispersing towards positive energies with increasing field. The observation of hole-like levels in the valence band is apparently hindered by their electronic disorder, as demonstrated in Fig. 5.1 c), and by their lower band velocity. The data also reveal that the spacing of the Landau levels decreases with Landau level index n, indicating a non-parabolic conduction band. Finally, satellite peaks for the dominant Landau level peaks are resolved at high field, revealing the lifting of a degeneracy with increasing field. Figure 5.3 b) shows individual spectra for the higher fields which resolve a double peak structure for up to the first 8 pairs of levels (e.g. 12 T). We first extract information about the band structure of Cd3 As2 from Landau level spectroscopy measurements using a model-independent method. The semi-classical Lifshitz-Onsager relation specifies that the extremal area Sn in reciprocal space for the Landau level n occurring at energy En must quantize as Sn = 2πe(n + γ)B/~, 75
a
c n=0
1
3
2
5
4
6
300
14
200
10
Energy (meV)
8 6 4 2 0 −250
dI/dV (ns)
−200
−150
0
1.2
−100
−50
0
Energy (mV)
b
2.4
50
100
k (1/A ) 0
−0.05
0.05
0.1
LL Index 0 1 2 3 ....
16
vF = 9.4 × 105 (m/ s)
100 0
n=4 n=3
−100
n=2 n=1
−200
n=0
−300 1
2 1
2
3
4 4
5
6
7
8
d
13.0 T 12.5 T
e Distance (nm)
13.5 T
12.0 T 11.5 T 11.0 T
f Distance (nm)
10.5 T 10.0 T
−300 −250 −200 −150 −100
Energy (mV)
−50
0
50
100
40
-200
0
-100
100 4
12.25 T
20 0
0
1
2 1
3
4
5
6
dI/dV (nS)
0
2
2
0
0 40
4
12.25 T
dI/dV (nS)
0
Distance (nm)
14.0 T
3
20
0 40
2
-20
0
20
12.75 T
20
0
0 4 dI/dV (nS)
Magnetic Field (T)
12
dI/dV (arb.unit)
−0.1
...
Fig. 2
2
-20
0
Energy (mV)
20
0
Figure 5.3: a) Landau level fan diagram measured at 400 mK, consisting of point spectra in 1 T increments. b) The point spectra, obtained from 10 T to 14 T in 0.5 T increments, show a doublet Landau peak structure whose separation decreases at high index. Plots are shifted vertically and a smooth background is subtracted based on the 2 T data. c) Effective band dispersion in the (112) plane formulated from the Lifshitz-Onsager quantization condition. Sixteen Landau levels for each magnetic field are plotted, where the average energy is used for indexes with two split peaks. d) Spatial variation of Landau Levels at 12.25 T. The green curve is the spatial average. e), f) Spatial variation of Landau Levels around the Fermi level at 12.25 T and 12.75 T. The spectra in d), e), and f) were all taken along the same line cut. 76
where γ is the phase offset of the quantum oscillations [121]. As verified by QPI measurements presented later, the constant energy contours in the (112) plane are nearly circular; hence, we can take Sn = πkn2 , where kn is the geometric mean of the high symmetric axes of the Fermi surface contour in the (112) plane. We use γ = 1/2 and adopt an intuitive assignment of the index n to the peaks, labeling every two with the same index starting with n = 0 as shown in Fig. 5.3 b). In Fig. 5.3 c), the average peak position En and its associated kn for various B fields trace out an effective dispersion relation. Remarkably, the entire set of peaks collapses onto a p single Dirac-like (n + γ)B ∝ |k| scaling for a wide energy range, revealing the strong linearity of the conduction band. The linear dispersion with very high Fermi velocity vF = 9.4±0.15·105 m/s extends to at least 0.5 V above EDirac , far beyond the expected Lifshitz transition where the two Dirac cones merge. While this extended linearity is not guaranteed by the Dirac physics around the band inversion, it presents important consequences for transport properties of samples with similar carrier concentration. For example, under the assumption of scattering from a screened Coulomb potential, 1/3
the mobility for a 3D linear dispersion scales as vF2 ne /ni , in stark contrast to the ne /(m∗2 ni ) scaling for a 3D quadratic dispersion, where m∗ is the effective mass, ne is the carrier density, and ni is the concentration of scattering centers1 . This contrasting physical regime for Cd3 As2 , which cannot be considered as the limit of normal band structures, may be critical to understanding the ultrahigh mobility and large magnetoresistance reported in a recent transport experiment [37]. Finally, we observe that the extrapolated crossing point from the high energy dispersion occurs at -300 mV, below EDirac , and that the effective velocities of the n = 0, 1 levels become increasingly small relative to the high energy behavior. We will explain below in detailed modeling that this deviation is a consequence of our sensitivity to the band minimum in the kz dispersion. 1 The relationship between linear and quadratic dispersions can be see via the substitution vF → k/m∗ and noting that ne ∝ k 1/3
77
5.4
Spatial Homogeneity of Landau Levels
Next, we discuss the spatial homogeneity of the Landau levels. In Fig. 5.3 d), we verify that the dominant peak positions are homogeneous in space, with exception of fine features which occur near the Fermi energy. In Fig. 5.3 e) and f), we show the n = 4 and n = 5 Landau levels for the respective fields when they approach and pass the Fermi level. Remarkably, in certain locations, we resolve a four-peak structure in the n = 5 level and weaker hints of splitting of the n = 4 state. Because this fine structure occurs in the vicinity of the Fermi level, we speculate that it may arise from band structure effects (states at different momenta but the same energy) that become resolvable near the Fermi level due to the extended electron lifetime, or from many body effects [120]. Since the four-fold structure shifts together with increased field as shown in Fig. 5.3 e) and f), we rule out half-filling of the Landau levels [11].2 As we are above the Lifshitz transition in this energy range, the additional splitting should also not be interpreted as the lifting of the valley degeneracy of the two Dirac points [11].
5.5
Quasiparticle Interference
Moreover, the spatial resolution of STM enables independent confirmation of the band structure derived from our Landau level spectroscopy measurements. The Fourier transform of spatial modulations in the local DOS mapped by STM provides information about quasiparticle interference (QPI) caused by elastic scattering wavevectors that connect points on the constant energy contour. QPI for a 3D band structure can be approximated as the integration of the 2D QPI intensities for the Fermi surface planes at fixed k3 perpendicular to the sample surface. Therefore, QPI for the nearly spherical constant energy surfaces of Cd3 As2 should represent the weighted sum of 2
Half-filled Landau levels splits an single Landau peak into two, resulting in peak on either side of the Fermi level. Hence it would generally result in an odd number of observed peaks.
78
a
dI/dV(pS)
501
817
b
dI/dV(pS)
90
272
c
10
g
55
-200 mV
150 mV
450 mV
dI/dV(pS)
600 500
150 A
d
dI/dV(pS)
0
2.7
450 mV
e 150 mV
dI/dV(pS)
0
0.7
f
dI/dV(pS)
-200 mV
0
0.1
Energy (mV)
400 300 200 100 0 −100
Landau levels QPI 6 band QPI second band
−200 −0.2
0.3 / A
0 k (1/A )
0.2
Figure 5.4: a), b), c) Spectroscopic maps of Cd3 As2 at 450 meV, 150 meV, and -200 meV, respectively. d), e), f) 2D discrete Fourier transforms (2D-DFTs) of a), b), and c), respectively. The red dashed circles show the scattering of the electronlike conduction band, and the cyan dashed circle shows that of a second band which emerges at higher energy. g) Plot of QPI peaks and reproduced Landau level peaks. The red (cyan) momentum vectors are obtained from the radius of the QPI feature. Blue and green curves are guides to the eye. The orange circles reproduce the Landau level data shown in Fig. 5.3 c). 2D QPI patterns for contours of “latitude” at fixed k3 . Each ring of latitude of radius k contributes a QPI ring of radius 2k; thus, the sum will fill up a disc-like pattern. The regions near the equator of the Fermi surface (maximal kmax ) will be weighted more due to the slower dispersion of the radius of the ring of latitude, while regions near the pole (k → 0) will be weighted less due to the faster dispersion. Consistently, the experimental QPI data contains significant contributions inside intensity up to the maximal radius 2kmax , as is exemplified for E = 150 mV shown in Fig. 5.4 e). In Fig. 5.4 a-c), we display spectroscopic maps measured at B = 0 T and T = 2 K for three different energies that display vivid wave-like features. The evolution of the QPI maps from E = 450 mV to E = 150 mV shows the length of the scattering wavevectors to increase with decreasing energy. At E = -200 mV, this interference signal can no longer be resolved as the diverging wavelength near EDirac overlaps with 79
the background electronic puddling. The interference patterns seen in the spectroscopic maps and their DFTs (see Fig. 5.4 d-f)) distinguish the shape of the extremal Fermi contour above the Lifshitz transition as quasi-circular, justifying our previous assumption. Figure 5.4 g) summarizes the extracted dispersion from the QPI and Landau level measurements, which together reinforce the consistent picture of a conduction band that onsets near -200 mV and disperses linearly at high energies. Above 500 mV, the linear dispersion becomes flatter and a second scattering vector, likely from another bulk band, is resolved in the QPI data.
5.6
Landau Level Simulation
To gain further insight into the nontrivial Landau level structure and to determine when Weyl fermions appear as the low energy excitations of Cd3 As2 , we introduce a band structure model that captures the salient features of our data. Following previous work, the low-energy dispersion around the point for Cd3 As2 can be described by an inverted HgTe-type band model using a minimal 4-band basis of the states S1/2 , 1/2 , P3/2 , 3/2 , S1/2 , −1/2 , P3/2 , −3/2 [41, 122]:
Ak+ 0 0 M (k) Ak− −M (k) 0 0 Hef f (k) = 0 (k) + 0 0 M (k) −Ak− 0 0 −Ak+ −M (k)
(5.1)
where 0 (k) and M (k) encode the band structure and k± = kx ± iky (details are described in Appendix D). Landau quantization in the (112) plane reflects both the kx -ky and kz dispersions, where the latter could not be precisely determined from photoemission measurements on samples with (001) cleavage planes [108, 109]. As these previous studies demonstrated linear kx -ky dispersion, the linearity in Fig. 5.3 80
Fig. 4
b
a
n=2
Data
Energy (mV)
0
n=0
•••
ELifschitz, lower
n=3 n=3
−50
n=1
40 30 20 10
n=2 n=2
−100
g* (∆En/μBB)
EDirac
0
30
60
90
θ from [001] (°)
d
n= 0 1 2 3 4 5 6 7
50
n=1
ELifschitz, upper
50
c
10.5 T 12.0 T 13.5 T
0 −200 −150 −100 −50
0
Energy (mV)
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50
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n=1
n=0
−200
Energy
n=0
Energy
−150
EDirac
EDirac
−250 0
2
4
6
8
10
12
14
[1 1 2]
B⊥(T)
Γ
k3
[1 1 2]
[0 0 1]
Γ
k3
[0 0 1]
Figure 5.5: a) Simulation of Landau levels and their splitting (peak positions of 10T to 14T Landau level spectra are plotted as red circles). The electron-like (blue curves) and hole-like levels (red curves) are derived from the extrema of the Landau level bands at the Γ point. The Dirac and Lifshitz points at zero field are marked on the vertical axis. b) Theoretical angle-dependent orbital splitting of the Landau levels. The measurements reported here were performed at =54.7, denoted by the yellow bar. c) Effective total g-factor g* extracted from the experimental data as a function of energy and magnetic field. d), e) Calculated Landau level bands for a magnetic field along the [112] direction and [001] direction. The corresponding calculated density of states is shown for the [112] directed field. The inset in (d) and (e) zooms in on the crossing point between the lowest electron- and hole-like bands, showing the opening of a gap in (d) due to broken C4 symmetry. c) implies that the kz dispersion is also linear at high energies. To capture this trend, we modify the original parabolic kz dispersion in M (k) to be hyperbolic. This simple modification maintains all qualitative aspects of the low energy band inversion and is essential for modeling the extended energy range of the data. When a magnetic field is applied, we transform the momentum k → k − e/~A via Peierls substitution of the magnetic vector potential A and include a Zeeman term in the total Hamiltonian H(k) = Hef f (k) + HZeeman (k). In Fig. 5.5 a), we show the results of numerical Landau level simulations using band structure parameters consistent with the kx −ky dispersion measured by photoe81
mission and with the presence of band inversion indicated by our zero field spectra. Although a precise determination of the size of the inversion is not possible (20 mV is used in Fig. 5.5), the data are more consistent with shallower band inversions. Nevertheless, the model illustrates the essential physical origin for the observed Landau level structure. At high fields, the DOS singularities observed in the data correspond to the energies of the Landau level band minima at the Γ point (see Appendix D for discussion of the low field regime where additional extrema may occur inside the two Dirac cones). Hence, the deviation from Dirac scaling for the lowest levels in Fig. 5.3 c) reflects the parabolic (massive) band minimum in the kz dispersion, which is probed by the tilted magnetic field. More importantly, the agreement of our data with the model calculations suggests that the Landau level doublet structure arises from a combination of orbital and Zeeman splitting of the spin-degenerate conduction band. Orbital splitting depends on the shape of the band structure and diminishes away from EDirac . In Fig. 5.5 b, we theoretically illustrate this evolution of the Landau levels due to orbital effects as the angle of the field is tilted away from the c axis (for clarity we have set the Zeeman term to zero here as it introduces only an additional nearly constant splitting). For our data, measured at the intermediate angle denoted by the yellow bar, it is natural to adopt the assignment scheme n shown on the right side of Fig. 5.5 b) such that the pairs of levels closest in energy have the same index. In Fig. 5.5 c), we extract an effective total g* from the experimental Landau level splitting for each index at several different magnetic fields. We find that g*=37±2 for the lowest level and that g* decreases with increasing energy from EDirac , consistent with theoretical models based on prior Shubnikov-de-Haas measurements [123].
82
5.7
Outlook
In the case of a magnetic field titled from the c-axis, calculations based on our model band structure show that the Weyl nodes are eliminated by small gaps at the Dirac points caused by the broken rotational (C4) symmetry (Fig. 5.5 d). Therefore, to observe Weyl fermions in Cd3 As2 , application of a magnetic field along [001] is required to break time reversal symmetry while maintaining C4 symmetry (Fig. 5.5 e). Moreover, the direction of the magnetic field is shown here to tune the orbital and orbital-independent splitting in this material. Exploration of that phase space in next-generation samples with lower carrier concentration opens the possibility of engineering and observing topological states in 3D Dirac materials.
83
Chapter 6 Conclusion This dissertation described the construction of a fully-featured, ultra-high vacuum STM capable of operating at 240 mK and at magnetic fields up to 14 T. Indeed, the rewards of this long effort have come into fruition through two experiments on unconventional heavy superconductivity in CeCoIn5 and the topological band structure of the Dirac semimetal Cd3 As2 . In the former experiment, strong parallels were established between (low temperature) superconductivity in CeCoIn5 and the high-Tc cuprate superconductors, most fundamentally in the dx2 −y2 symmetry of Cooper pairing. Moreover, the appearance of a pseudogap independent of, yet in close proximity to, the superconducting gap in CeCoIn5 represented another commonality between this heavy fermion superconductor and the cuprates, where such commonalities may be expected from the similar phase diagrams dominated by the presence of antiferromagnetism. In the latter experiment, the magnetic field was used to resolve the astonishing dispersion of the conduction band of Cd3 As2 , linear in three momentum dimensions over nearly a volt range and with two-fold spin degeneracy split by a g-factor as high as ∼40. Naturally, the performance of this singular instrument has motivated a long list of future experiments, both to extend and answer the questions raised by the initial CeCoIn5 and Cd3 As2 experiments and to span new fields, such 84
as single spin dynamics in semiconductors and the quest for Majorana fermions in 1D superconductor-nanowire systems. As the list is as endless as one’s curiosity, this conclusion will modestly only address future avenues in heavy fermion and topological semimetal research. The demonstration of the close relation between superconductivity in CeCoIn5 and the high-Tc cuprates opens a new avenue to understanding magnetically-mediated superconductivity. Moreover, this new avenue can even be advantageous as the stoichiometric CeCoIn5 samples are ultra-clean, with coherence lengths ∼50 ˚ A, three to four times longer than that in the cuprates, enabling real space effects visualized by STM such as vortex physics and impurity quasiparticle states to appear particularly striking as we have shown. The natural first order of business should be to explore the phase diagram of CeCoIn5 through doping. Hole doping with Cd (equivalently Hg) on the In sites has been shown to induce antiferromagnetism (AF) - above 0.75% Cd, AF coexists with superconductivity, which gives way to only AF above 2% Cd. The QPI and impurity state experiments should be repeated both in the coexistence regime and in the purely AF regime to contrast with the results of undoped CeCoIn5 (pure superconductivity). Does the size or depth of the pseudogap scale with the Neel temperature TN , which would connect the pseudogap to magnetic fluctuations if so? It is further believed from nuclear magnetic resonance studies that AF nucleates locally around single Cd impurities before establishing long range order. The STM can ideally address this hypothesis by tracking the local spectroscopic signature around Cd impurities as a function of the density of Cd and imaging the nucleation and growth of this transition. Moreover, electron doping CeCoIn5 , such as with Sn for In, or doping onto the f -electron site, such as with Yb for Ce, would also be interesting. Both these types of dopants (particularly Yb) are believed to less drastically impact superconductivity than Cd does; hence, understanding the microscopic rea-
85
son for this may have implications for engineering more robust, higher temperature magnetically-mediated superconductors.
Figure 6.1: The interplay of doping and superconductivity in CeCoIn5 . What happens on the local level when hole doping with Cd introduces AFM, while electron doping with Sn much less strongly suppress superconductivity? Figure courtesy Eric Bauer. Finally, studying CeCoIn5 in non-natural cleaveage planes, such as (100) plane for example, would enable application of the magnetic field in the ab plane of the sample, due to our experimental restriction of being able to only apply the magnetic field along the sample normal. A (100)-oriented magnetic field would allow access to exotic high field phases in CeCoIn5 , such as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase (spatially modulated order parameter with Cooper pairs between |k, ↑i and |−k + q, ↓i, comprising a center of gravity of finite linear momentum) and the so-called “Q” phase (superconductivity coexisting with a spin density wave). What are the spectroscopic signatures of such phases? Generically, a non-spin polarized tip is not sensitive to the spin channel; however, perhaps spin modes may couple to the superconductivity creating a ‘wiggle’ in the spectrum outside the gap at E = ∆ + ω similar to electron-phonon coupling seen in BCS superconductors by McMillan 86
and Rowell [124]. Alternatively, both phases predict some spatially modulated gap strength; however, very careful data acquisition and analysis will be necessary to isolate the true effect on the already much diminished gap strength (due to the high magnetic field) from the unrelated, but coincident effects of vortex physics and STM setpoint. These are difficult, but pioneering experiments at the frontiers of this heavy fermion superconductivity field.1 With regards to Cd3 As2 , our experiments demonstrated that the band inversion that forms the Weyl nodes is shallow, likely < 50 mV. Although this may seem small, in actuality it is quite reasonable since even 25 mV corresponds to approximately room temperature, 273 K. The obstacle, therefore, in Cd3 As2 is a materials one that the Dirac point, predicted to be exactly at the Fermi level for the stoichiometric compound, is nevertheless buried by 200 mV below the Fermi energy by native defects introduced during imperfect growth, particularly As vacancies. The buried Dirac point and its inhomogeneity in space introduce strong lifetime and disorder broadening of the lowest, chiral Landau level, and therefore mask the nontrivial effects due to it. To raise the Dirac point up to the chemical potential is a challenge for our chemistry colleagues and will likely require less defected, intrinsic samples, rather than the current (intrinsically) electron-doped samples to be compensated with hole dopants, which would introduce more disorder. Another avenue, long term, may be to pursue thin film samples grown by molecular beam epitaxy, where the addition of a back gate may be used to tune the carrier concentration. Very recently, Ref. [126] reported ARPES measurements on Cd3 As2 samples with Fermi energies pinned to the Dirac point, where such low carrier concentration samples have been traditionally very hard to synthesize.2 However, this paper also claimed the absence of Cd 1
Very recently (Dec. 2013), Ref. [125] discovered the Q-phase in Ce0.95 Nd0.05 In5 in zero magnetic field (and coexisting with superconductivity (Tc = 1.85 K, TN = 0.9 K)). This would enable the study of the Q-phase on any cleavage plane of CeCoIn5 and absent of the effects of vortex physics. 2 This is at least my impression from reading papers from the 1970s, the last period of strong basic research in Cd3 As2 . Admittedly, growth techniques may have advanced significantly in the intervening 40 years.
87
ordering (random Cd vacancies) in the surface layers, attributed to diffusion after the cleaving procedure. Our atomic resolution STM images show any reconstruction or supercell of the surface atoms to be regular (at least over O(50 nm) sized patches); hence, suggesting that the Cd ordering on or near the surface is periodic, rather than random. However, the supercell, although similar, is difficult to precisely match 1to-1 with the atomic structure presented in [114] for the bulk of Cd3 As2 . Thus, in the fast moving field of topological semimetals, it perhaps prudent to take a breath to carefully resolve the nuances of the crystal structure and growth, as the properties of the band structure hinge directly on these basic assumptions.
290 x 290 Å
54 x 54 Å
Unit Cell: 16 atoms; 4 displaced
Figure 6.2: One of the developing controversies in study of Cd3 As2 is the direction of the Dirac points on the (112) cleaved surface, which is determined by the symmetries of the crystal lattice. In Ref. [126], it was claimed that upon cleaving, Cd atoms diffuse in the surface layers to randomize their ordering pattern and therefore shift the Dirac points from the bulk [001] axis to the [112] axis. Our STM measurements on cleaved Cd3 As2 show a zigzag ordering of the surface atoms likely reflective of the Cd ordering, either on the surface layer (if this surface layer is Cd) or in the layer underneath (if this surface layer is As). Regardless, this pattern has a regular supercell of 16 atoms, with 4 atoms significantly distorted from their ideal hex lattice position. This fraction of distortion (25%) strongly suggests it is reflective of the Cd ordering (25% vacancy). Hence, our measurements suggest that the Cd is indeed ordered on the surface layers. Once the materials issues are addressed, further STM studies should be performed to reveal both the electron- and hole-like Landau levels and pin down the details near the Dirac point. Study of both the (112) cleave, discussed in this thesis, and the (001) 88
cleave, as claimed by ARPES experiments, will allow the search for the topological effects predicted for this Dirac material. For the (112) plane, Fermi arc surface states, with long wavelength due to the small separation between the Dirac points, may be revealed once the background charge inhomogeneity is reduced. For the (001) plane, where applying a magnetic field preserves the C4 symmetry, the addition of an electric field from a back gate may enable the search for E · B chiral current effects. In any case, the ability to study two cleavage planes necessarily tunes the direction of the magnetic field and it may be worthwhile to verify the angular dependence of the Landau level splitting presented in Fig. 5.5 to fully justify the underlying model. In conclusion, a promising future for materials research at high fields and ultra-low temperatures awaits. However, the technical advance achieved in this dissertation should be considered only the first step in a continual effort to push the boundaries of experimental capability. Indeed, plans are already underway to design a new dilution insert that can be substituted into the cryogenic and vibration support structures of this current instrument and that upgrades upon its performance and convenience. The capability to run continuous measurements with a thermally anchored fixed impedance to continuously feed the 1 K pot would drastically change the measurement duty cycle of the instrument. Moreover, careful planning and implementation of multi-stage RF filtering for all electrical lines down to the microscope will aim for lower electron temperature. The STM landscape is rapidly evolving as in the next decade, high performance dilution refrigerator-based systems will become commonplace, even commercially available. Hence, continual innovation at labs such as Princeton will be required to push and stay at the forefront of the technique.
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Appendix A Further Experimental Aspects of DRSTM This appendix describes further details of DRSTM operation and design to support the more general, global overview given in Chapter 3.
A.1
X and Z Capacitance Position Sensors
The two sample design of DRSTM is a wonderful institution in that if the tip ever ‘fails’, due to acquiring a bad tip shape or an insulating spectra, the operator can simply walk over to the auxiliary Cu sample, reprep the tip by field emission, and come back to the sample under study with a fresh tip. However, on samples that do not cleave well, such as in particular the CeCoIn5 samples studied in this thesis, it is exceedingly rare to approach a large atomically terraced area and hence the advantage of the two sample holder design would be somewhat minimized if the tip is damaged reapproaching ‘bad’ areas.1 Hence it is imperative that once an decent 1
Usually a fresh tip can withstand a few (90%.
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region is located, its location be tracked in situ with a sensor of the x-walker position. Here, I describe an innovative, small form factor differential capacitance sensor that crucially maintains high sensitivity for the x-walker position on both the left and right extremes of its motion.
a)
c) L CL
CL
CR
CR
VL R
5
L
d)
Data Fit
5.2
L
R
0 −5 −10 −15 0
R 2 4 X Position (mm)
CZ
R2
R1
10
C −C (uV)
VR
Z Capacitance (V)
b)
Ctip/sample
Vosc
5
4.8 Approach (45 mV/1K) Retract (65 mV/1K)
4.6 0
6
5 10 Steps (Thousands)
15
Figure A.1: “The Eyes of the STM”: Capacitance sensors for X (a) and Z (c) Pan Walkers. (b) The X walker metric is a differential lock-in measurement performed at 617 Hz with an oscillator amplitude of 0.25 V (0.5 V into the divide by 2 box we use conventionally; R1 ∼ 5 kOhm). (d) The Z walker metric is performed applying the same 0.5 V oscillator to one of the z walker capacitance electrodes, amplifying the current pickup on the other electrode with an Ithaco preamp at 10− 5 gain, and reading the voltage on the lock-in at 1 mV sensitivity. Note that the inner metal part of the z capacitance sensor has a ∼10 kOhm short to ground, which fortunately only negligibly impacts the measurement as the wire resistance is ∼100 Ohm. The principle of the x-walker capacitance sensor is shown in Fig. A.1 a). Two additional floating pieces of metal (shown as the orange L’s) are placed on the inside of the sample carriage arms (shown in blue). The orange pieces are fixed to the microscope body and do not move, while the blue arms are translated by connection to the sapphire prism on the horizontal Pan x-walker (above). Hence, the separation 91
of the blue arms is constant, and the arms move relative to the orange pieces. Thus, a metric of the position of the walker is the difference in capacitance CL − CR , where CL (CR ) is the capacitance between the left (right) sample arm and the left (right) capacitance plate. Notice that critically this metric maintains high sensitivity when the walker is close to either the left and right sides (where the tip is over the samples), but is less sensitive in the middle (where conveniently this is the gap between the two samples). Readout of this capacitance is performed via lock-in detection through the bridge circuit shown, where an oscillator is applied to the sample arms (i.e., the bias of the STM) and the voltage pickup VL and VR are differentially detected by the lock-in (VL − VR ∝ CL − CR by simple analysis), thus canceling common mode noise. One of the two resistors R1 can be tuned to cancel any slight offset such that the reading is 0 V when the walker is exactly centered in its range. In Fig. A.1 b), we show the experimental capacitance trace as the walker is moved over (the greater part) of its ∼7 mm range of motion. The red line is a fit to the equation: C(x) = C0
1 1 − , x x0 − x
(A.1)
representing the difference between two parallel place capacitors. We note that the instantaneous error in the capacitance reading is ±0.003 uV, but the reproducibility (backlash) after walking away and back is probably of order 0.02 uV. Hence, near the center of the samples,2 the backlash error can be converted to a effective positional reproducibility of order 5 um. In addition to the x capacitance sensor, DRSTM possesses a linear z capacitance motion sensor, which was useful during the testing of the walking of the z-walker in a 4K dipstick, however in daily operation now has been supplanted by measuring the 2 For historical purpose, we note that the CeCoIn5 approach that engender the published QPI and impurity maps was at x = +5.83 uV.
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tip-sample capacitance, as will be described in the following section. As shown in Fig. A.1 c), the z-capacitance sensor design consists of two concentric cylinders, the inner cylinder which is attached to the bottom of the scanner tube (which is walked by the Z Pan walker) and another outer cylinder which is stationary. Testing of the z-walker at room temperature using a Nanonis waveform of amplitude 125 V (hence bipolar peak-to-peak of 250 V) results in a linear position signal which is shown in Fig. A.1 d). The important aspect of the figure is the ratio of the backward (retract, with gravity) to forward (approach, against gravity) speed (= 65/45 ∼ 1.5) which is a good diagnosis of how tightly the Pan walker should be tuned at room temperature. If this ratio is too large, the walker (BeCu spring plate) is too loose and walking up is inefficient; if this ratio is too small, the walker is too tight and may seize at low temperature.
A.2
Life on DR: Including Approaching an Sample
After the sample has been demonstrated to possess well-cleaved areas, the experimental lifetime is essentially infinite on DRSTM. The same area of the sample can be tracked from 0 to 14 T. In some situations, a few steps must be taken in z, in order to keep the sample at the safe z-range for refilling. However, the trained operator can easily track the same area even with walking a few steps due to the reliability of the walkers. Hence, the measurement will continue until the user is unhappy with the tip, at which point the microscope is moved to field emission on copper, and afterwards moved back over the sample using the same registry of the x-capacitance sensor, described above. The next step is to approach the sample, that is to say, to safely position the tip within 10 ˚ A of the surface, starting from a macroscopic millimeters away, without ever having the tip be in direct contact with the sample. This of course is the whole marvel of STM and its solution relies on the utilization 93
of piezoelectrics. With a combination of a piezoelectric coarse stepper motor and a piezo tube scanner, the problem is reduced to using the stepper motor to place the STM tip within the range of the piezo tube scanner (∼150 nm). In principle, this can be achieved by checking for the tunneling current after each step of the motor, if the range of the piezo tube scanner (how much its length changes given the full range of applied voltage) is larger than the step size. In ‘safe approach’ or ‘tip retract mode approach’, the tip is first retracted (compressed). Then a step is taken, and the tip is released slowly (extended) to check for tunneling current, which as soon as it finds it, the tip is retracted and no more steps are taken. Since the motor step is smaller than the range of the tube scanner, this checking for tunneling current is redundant, and the sample surface will be found before any step towards it crashes the retracted tip into the sample.
Tip/Sample Reading (mV)
78 76
117 Data Approached Linear Background after 656 steps Background + Sample
Start “safe’ approach
74
114 111 108
72
106
70
103
68
100
66
97
64 62 0
True Capacitance (fF)
80
94
Commence approach after FE 5
10 15 Z Steps (Thousands)
91 20
Figure A.2: The tip-sample capacitance versus steps taken during the approach. The experimental trace is fit to the empirical form Ctip/sample = C0 + C1 x + C2 /(x0 − x). The total approach took ∼ 20,000 steps, where only the only during last 656 did we check for the tunneling current after each step. It is intriguing that the tip-sample capacitance in real units is 100 fF when the sharp metallic tip is within ∼100 nm (fully retracted scanner tube) of the sample (infinite plane). Generally the first part of this process (moving to, performing, and moving from the field emission) takes about 2 hours. Thereafter, the approach can be easily done 94
in under 90 minutes if the tip-sample approach capacitance (the capacitance when we are within the range of the scanner piezo) is known. This value is actually relatively insensitive to the sample since the sample always looks like an infinite plane, but is more sensitive to the z-height (how thick of a sample holder you use) due to background geometric factors.3 Here we apply a oscillator on top of the bias of the sample4 and measure the resulting capacitative pickup on the tip using the same current preamplifier for the tunneling current. Of course, the ‘safe approach’ mode is extremely time consuming compared to simply taking steps without checking the current, and one only launches ’safe approach’ mode once one is sufficiently close to the sample. To know when we are sufficiently close to the sample, we monitor the tip-sample capacitance, which is displayed for a prototypical approach as shown in Fig. A.2. Empirically the Ctip/sample is linear far away from the sample (we can consider this linear term as background, or as a geometric effect from the sides of the tip rather than the apex), but near the sample it is dominated by a 1/z divergence. Safe approach is only launched when the derivative in Ctip/sample exceeds a certain threshold. In Fig. A.2, we required 656 steps after launching safe approach, which at a frequency of ∼11 steps per minute for settings that I like to use, took almost exactly 1 hour of approaching. This is a reasonable, if not conservative time, and the daring or impatient can reduce even further.
A.3
‘Joule-Thomson’ 2K Mode Operation
One of the downsides of the current operation of DRSTM is the necessity to single shot the 1K pot for optimal noise performance, resulting in a run time around 8 hours. As a result, conductance maps are taken one energy at a time, which can be frustrating if the tip changes in between. Oftentimes it is more useful to have the complete set 3 4
May also be slightly sensitive to x-location of approach. Generally we apply, 1V divided by 20 = 50 mV at a frequency of 434.3 Hz.
95
of energies over a partial area of the map (if the tip has to change), rather than the incomplete set of energies, but a few energies finished completely, over the full area. This was precisely the scenario during the high field Bi2 Sr2 CaCu2 O8+δ experiments completed during the summer of 2013. The necessity to run 72 hour maps during that time spurred operation of the system in so-called ‘Joule-Thomson’ (JT) mode, where the pot is cooled from a completely empty state and the needle valve is cracked incrementally until the 1K pot barely cools to a minimum of ∼2.1 K. If the NV is opened further (T1K ∼ 2.0 K), then liquid will begin to accumulate inside the belly of the pot and noise performance will slowly deteriorate as the liquid level rises. It the pot is not started from a completely empty state, the influx of liquid corresponding to the same NV setting as JT mode would also introduce excessive noise to the the turbulence of the mixing of the existing superfluid liquid inside the belly with the influx of ∼4 K liquid. Hence, it is imperative that the belly of the pot be empty and the incoming liquid be just enough to cool the pot through evaporative cooling as it turns to gas.5 Likely, the gas expansion theory is correct for JT mode operation between 3 K and above. For the lower temperatures near 2.1 K, it is believed that the influx of liquid is only enough to fill up the smaller capillary tube that feeds the pot and wraps around the bottom of the pot, leaving the bulk of the pot still empty. Here, we may have some 1K pot action, but for an extremely small diameter tube, rather than for the pot directly, and is empirically determined to be less noisy. The key point is that once JT mode is established, it is continuous and the system run is limited only be the refill of the main dewar, which is 3 and a half days. Moreover, a further advantage of JT mode is relative absence of pot ‘pings’. Fig. A.3 a) shows the initialization of JT mode from an empty pot state. Each additional cracking open of the pot results in approximately 0.3 K cooling. Notice that if the system had liquid in the pot, each opening of the NV would result in 5 Hence, the analogy to Joule-Thomson cooling, where a closed cycle of expansion of pressurized gas produces the cooling power. Here we expand from atmosphere to vacuum.
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a)
b) NV 2.8 NV 2.9
T 1K Pot
NV 3.0
T MxChb
Figure A.3: a) Engaging Joule-Thomson mode by slightly cracking the needle valve. The mixing chamber temperature has not had enough time to equilibrate. b) JouleThomson mode closed feedback noise performance for a setpoint of 100 pA at +400 mV on Cd3 As2 . warming of the pot temperature, since the pot has to cool the incoming liquid. This trace shows that we can in principle stabilize at all temperatures between ∼10 K and 2.1 K in approximately 0.5 K increments. In general, the temperature will drift once the main dewar liquid level falls below 25%. The closed feedback noise performance is displayed in Fig. A.3 b) for a setpoint current of 100 pA at +400 mV on Cd3 As2 , where all QPI maps on Cd3 As2 for were taken in this mode. Comparison to the performance of single shot pot, fridge not running (Fig. 3.7 b) orange trace6 ), shows worsened noise at frequencies between 850 Hz and 1250 Hz; however, the noise is still much better than single shot running the fridge, hence is quite capable of taking excellent data in conjunction with the multipass technique.
A.4
Dewar Exhaust Management
Figure A.4 diagrams the connection of our main bath boil-off to the Princeton recovery system. Proper management of this boil-off is necessary to allow the dewar to stabilize after refill, prevent dewar ‘pings’, and reduce the sensitivity of our system 6
The trace in Fig. 3.7 is open feedback, but the feedback loop does not affect frequencies above ∼200 Hz
97
to fluctuations in the recovery back-pressure. The boiloff from the main bath is connected out of the dewar through its most central port (thus believed to better guide the boiloff through the vapor shields on the insert) with a rubber hose with NW16 tube fittings. Rubber is chosen for its flexibility in order to minimize transfer of vibrations to the granite table, and the small diameter tube chosen in order to minimize the transfer of acoustic noise into the dewar. The NW16 rubber hose is adapted to NW 40 PVC hose at a concrete block (in which all pumping lines are cast) sitting at the edge of the floating plinth. Thereafter, the boiloff runs through a Hayward PVC ball
V
Vent
V
To Building Recovery
V
Foam
V
V
To Yazdani Bag
Alicat PD Pressure Controller
NW40 everywhere else
Hayward Ball Check Valve
NW16 Rubber Hose
Main Bath
Figure A.4: Flow chart for the boil-off from the experimental main bath. The ’V’ symbol denotes ball-valves that can be either opened or closed. The Alicat pressure controller is model PCR-30PSIA-D/5P LIN,5IN Range:800Torr. check valve, which defends against back-flow. A piece of polyurethane foam is stuck in the line to block acoustic noise from entering the dewar, as the exhaust lines reach the recovery inside the noisy pump room. Together the foam and Hayward check valve “constipate” the line, increasing the time constant for any pressure changes in the main bath which is good for stability. The boiloff then continues to an Alicat pressure controller which controls the pressure in the line. The Alicat is generally set for a setpoint of 780 Torr, with proportional gain 15000, and differential gain 1500. The gain should not be set so high as to cause “thudding” of the pressure controller. During quiet measurement, the boiloff is routed after the Alicat to Yazdani lab’s ded98
icated bag, which has its own compressor and acts as a buffer before going into the building recovery. Connecting directly into the building recovery is problematic, as the building recovery’s pressure is sometimes higher than the setpoint 780 Torr and is highly sensitive to refills that occur in other labs. A series of bypass valves is in place to enable a straight shot to atmosphere or recovery during high dewar pressure situations, such as during refill and during the movement of the insert up and down for sample transfer.
A.5
Electrical Ground Loop Management
Single Grey Power Outlet
Computer
Single White Power Outlet Preamp strap
Extremely Thick Copper Braid
Ion Pump ‘Deadman’ Controllers
Preamp
Grounding Bar
Ion Pump Chasis Ground
Rack System (Insert and Chambers)
IGH Black Power 208 V 3-Phase
Rack ground through cable shields, etc.
Figure A.5: Schematic for the electrical wiring of the DRSTM instrument. The key is to have all power touching the system (preamp, ion pump, Nanonis controller, other measurement electronics) emanate from a single white power outlet. While certainly not as deleterious to STM performance as vibrational noise, which scales with the tunneling current, minimizing noise from 60 Hz and its harmonics due to electrical ground loops is a workman-esque way to gain the slightest bit of an advantage. While certain principles, such as cutting ground loops when possible and 99
minimizing its length and resistance when not) can guide in the reduction in 60 Hz noise, the ultimate test is empirical: try out various configurations and settle on the one with the cleanest tip-retracted noise spectrum (out of tunneling range). Figure A.5 shows the configuration of the DRSTM electronics which produced the best 60 Hz characteristics. In measurement condition, the only electronics touching the system that must be powered are the Femto preamplifier and the ion pumps. We make sure that these are supplied with white (filtered and backed up) power from the same power outlet. Furthermore to eliminate the ground loop to the ion pump, the ground plug of the power outlet is broken to the ion pump via a “deadman” connector (i.e., the ion pump only gets neutral and hot). Instead, chassis ground for the ion pump is provided directly from the chambers via a ground strap (star distribution of ground with the chamber at its center). The chambers and insert themselves are grounded to the copper grounding bar at the back of the room with the thickest copper braid that we could purchase. We then wire power to the electronics rack, containing Nanonis SPM controller, SRS lock-in, oscilloscope, helium level meter, from the same white power outlet. Additionally, the rack obtains ground from the insert via the shields of the cabling (scanner, walker, etc.) that touch both the insert and rack. One would think that one should use the same white power for the computer that communicates with the SPM controller; however, the computer is actually quite noisy and having it off white power, on its own separate “grey” power was more advantageous. The remaining electrical parts that must touch the system are the thermometer cabling and needle valve control cabling from the Oxford IGH, which is powered from a mixture of black power and 208V three phase power. The thermometer cabling is not an issue. Although minimal, the needle valve motor represents the biggest source of 60 Hz noise for our system, and we oftentimes manually disconnect it during measurement (although likely does not matter). Finally all metallic pumping lines 100
that touch the system are electrically broken somewhere farther away from the insert with plastic o-rings and plastic clamps (we don’t do this at the insert, since it would represent a ‘hole’ for electromagnetic radiation to enter and heat the inside of the system). Finally, to solidify the BNC ground connection between the preamp and the insert, a strap is used to clamp the outside of the BNC on the preamp to the barrel of the insert port.
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Appendix B Multipass Spectroscopy: An Alternative to Conventional Conductance Spectroscopy In this appendix, I describe a novel technique developed for acquiring the energyresolved conductance maps in spectroscopic imaging STM. The so-called multipass technique is in principle equivalent to the traditional method of conductance maps, but has been implemented in DRSTM for both the CeCoIn5 and Cd3 As2 experiments for its better noise performance and time efficiency when taking single energies.
B.1
Traditional Conductance Maps
Figure B.1 delineates the measurement cycle for obtaining the energy-resolved spatial density of states maps using the traditional method, called “conductance mapping”. Here, one can think of the individual measurement unit as the ‘point’ in that a series of N 2 repeated measurements must be made at each point on a N by N real space grid. At each point, first the setpoint condition (the particular height of the tip z that establishes the setpoint tunneling current at a particular setpoint voltage) is reached 102
and then the feedback loop is opened and the bias is ramped over the increments of voltage, recording the differential conductance for each specified voltage. Crucially here, one must wait at least three times the lock-in time constant Tlockin between each voltage, otherwise the history from the previous voltage will affect the next voltage’s measurement. In general, the traditional method is rather ‘jerky’ (i.e., stop and go between each point, between each voltage). In other words, the fundamental measurement outcome that is analyzed - the conductance at a particular energy over the entire field of view - is not taken in a continuous manner, but must be reconstituted over measurements separated in time. This technique therefore randomizes the effect of noise that is coherent over short time scales (the usual damped vibrational resonance oscillation), distributing it between the different energies. As a consequence, the FFTs of the conductance maps contain “white noise” or “salt-and-pepper” noise, which is spread throughout Q-space.
Figure B.1: Tradtional conductance map duty cycle. A ramped dI/dV measurement is performed at each point on a N by N grid, with a delay between each voltage to allow the lock-in to settle. Schematic courtesy Kenjiro Gomes.
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B.2
The Multpass Technique
In contrast to traditional conductance mapping, where the measurement unit is the point, the measurement unit for the multipass technique is the “line”. The essence of this fundamental restructuring of the total spectroscopic imaging measurement is the following: the traditional conductance mapping technique prioritizes continuity in energy, while the multipass technique prioritizes continuity in space. For a particular energy, we think of taking the individual N lines of a N by N grid one at a time, where in general adding to the number of pixels of the line (e.g. taking a N by M grid) does not necessarily affect the total measurement time. For each energy, a new pass over the line must in principle be accomplished. Historically, the multipass technique was developed under conditions where the measurement time was limited by the singleshot hold time of the 1 K pot (at base temperature), such that only a single energy of the conductance map could be taken at once using the traditional method. For simplicity, let us first describe the multipass technique for a single energy. FWD1: Scan the line at setpoint conditions with feedback closed at T1 = M * (Time per Pixel). Record line of setpoint heights z. BWD1: Scan back to start of line at TB, some fast speed ~ 1s. FWD2: Ramp bias to V1. “Play back” recorded z signal from FWD1 with same time T1 and take simultaneous dI/dV data. BWD2: Scan back to start of line at TB, some fast speed ~ 1s. (Technically feedback is also off here for bias ramping. We actually play BWD2.) Repeat for all energies before moving on to next line. On/Closed
Feedback
Off/Open
Z Position VSetpoint
Bias (V) V1 FWD1
BWD1FWD2
BWD2 FWD1
V2 BWD1FWD2
BWD2 Time
Grid N lines by M pixels per line
Energy V1
Energy V2
Figure B.2: The multipass techique divides the grid into a series of lines. The setpoint values for an particular line is first recorded with the feedback loop engaged. The bias voltage is ramped, and the tip is then asked to retrace the same line with the recorded setpoints while the feedback loop is disengaged. The simultaneous dI/dV due to a small summed oscillator is measured.
104
In the multipass technique, the tip is first scanned over a line (slowly) at the setpoint condition, and the z-positions (i.e. setpoint values) for the entire M pixels of the line are recorded and memorized. The tip is then rastered back (quickly) to the original starting point of the line, and the feedback loop is opened. At this point, the bias is ramped to the particular energy of interest, and the tip is asked to retrace the memorized z-positions over the same line with the feedback open. This is indeed a blind retracing and hence requires some stability of the STM instrument, but we have found it to be no more stringent than that demanded by the traditional conductance map, as in general line times are of order 20s, and this is a normal time for the feedback loop to be open for spectroscopic dI/dV measurements. But to reemphasize, here the feedback being open does not mean the tip is stationary, but rather than the tip is asked to retrace a particular x, y, z line a second time without regard to the setpoint current. During this retracing line, the small oscillator voltage is turned on, and we spend on average 6 to 8 lock-in time constants per pixel such that the lock-in records the simultaneous differential conductance as the line is traced. If faster settings are used, then the lock-in effectively performs a hardware Gaussian-smooth on the conductance map, where this is already common practice in post-processing of maps taken by the traditional method.1 In some sense, the multipass technique is similar to the method of simultaneous dI/dV mapping with topography; however, the crucial difference is that for the former method, we are able to use the same setpoint condition (i.e the same setpoint voltage V0 ) for all energies because of the adaptation of tracing a line feed-back open, rather than using using a different setpoint voltage for each energy map for the latter method. In this sense, in the absence of drift, the multipass technique should perfectly simulate the 1
For the CeCoIn5 experiments, we used 28 s per line over 272 pixels with a 10 mS lock-in time constant. Here the oscillator was 66 uV (very small) hence required longer averaging. For Cd3 As2 , we used faster settings at 14.9s per line over 208 pixels with 10 mS time constant and a whopping 7 mV oscillator. Interestingly, the two experiments performed in this thesis span the range of the fastest dispersion (heavy fermion) to slowest dispersion (light-like), as reflected by the range of oscillator amplitudes used.
105
conductance map for a particular energy taken by the traditional technique. The described technique can be implemented using the “record and play” feature of the “multipass” module of the Nanonis control system, calling a user written VI for greater convenience and for bias ramping, rather large single-stepped bias changes.2 Let us now discuss the pros and cons of the multipass technique. For a single energy, the main pro is that the bias is not ramped at each point (thus you don’t need to wait the lock-in settling time at each point). The bias is only ramped once at the start of the line! The lock-in settling time thus can be used as integration time, enabling much faster, or quieter maps. This is the most critical benefit. A second benefit is the that noise correlated on short time scales is now imprinted onto the image as in general, smooth coherent background modulations (of a set wavelength and direction depending on the line by line timing). Of course, the scale of the noise that survives the integration time should be much weaker than the signal of interest and can be made at a different frequency that the real waves by suitable choice of timing. Effectively, this concentrates the effect of the noise into a single peak or region of the FFT, instead of having it as white noise over the entire FFT (due to its more stochastic nature if it is distributed among different energies for the traditional method, resulting in lack of spatial coherence). The idea is to keep the noise coherent in space rather than white, such that it can be effectively filtered by the FFT. Another trick one can then play is to set the integration time per pixel to an exact integer multiple of the dominant noise period.3 The con of the multipass technique described above is that each energy is taken separately and hence the maps at different energies will not line up as perfectly (the delay between energies is on order of 10 s rather than 0.1 s). Also generally, we need to take the full set of energies with the same tip, and you run the risk of having 2
There are some details here relating to writing your own VI, in particular to how bias changes are handled depending on whether the feedback loop is on or off for a “record” (on) or “play” (off) line that the operator should be careful of. 3 The so-called ‘Delft’ technique courtesy Stevan Nadj-Perge.
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the tip change in between energies and not being able to crop the full map down in size to salvage the set of data. The extension of the multipass technique to multiple biases was implemented during the high field Bi2212 experiments. The conceptual extension is simple, after finishing a particular line at one bias, just repeat the same line at a different bias, and only move onto the next line when all biases are finished. The question then becomes whether to use the same recorded setpoint trace for the second/or third bias, or to re-record the setpoint line again. The author chose to be conservative and recorded the setpoint line each time for each different bias.4 This multiple recording of the setpoint z for each bias negates the time advantage (from not having to wait lock-in settle at every point) of the multipass technique when the number of energies becomes large. However, even on Bi2212, much higher quality data was obtained with multipass technique than with the regular conductance mapping technique for the same overall amount of time, testifying to the better noise performance. However, a perhaps crucial con is that the multipass technique requires tracing over the same area many times (4*NBias if fact), where the traditional technique traces over it twice (forward and back). On hostile materials such as Bi2212, the more you scan an area the more likely the tip is to change.
B.3
Conclusion
In summary, we have developed an alternative to the traditional conductance mapping technique that has its advantages in particular when very high quality data is needed for a small set of energies. Most importantly, the ability afforded by this technique to isolate noise in Q-space makes high quality data possible even in intrinsically more difficult systems to operate, such as the dilution fridge system in this thesis.
4
With more advanced drift correction code, its very likely the setpoint line can be predicted for later lines, rather than measured again.
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Appendix C Comparison of QPI in CeCoIn5 to Other Band Structure Probes and Phenomenological Modeling This appendix discusses further details of the quasiparticle interference patterns measured in CeCoIn5 , whose rather complex three dimensional band structure necessitate particular care in interpreting the data.
C.1
Reference to Other Experimental Mappings of the Band Structure
Since QPI measures the momentum transfer vectors (Q) which connect two points on the Fermi surface (FS), rather than the k-vectors of the FS directly, inferring a unique FS from QPI in a three-dimensional, multi-band material without making a large number of assumptions is not possible. As schematically illustrated in Fig. C.1, three bands cross the Fermi level of CeCoIn5 (identified previously by various theoretical and experimental efforts, including quantum oscillation and ARPES): band 108
135 (i.e., “α band” with cylindrical Fermi surfaces around the M points), band 133 (i.e., “β band” with a large, complicated Fermi surface), and finally band 131 with small Fermi surfaces [127, 85, 86]. Because of the expected light mass for band 131, inconsistent with the rapid dispersions seen in our Q vectors, and its small size, we focus instead on the α and β bands for the origin of our Q vectors (specifically to only the surfaces of α and β seen in quantum oscillation experiments). The decrease in length of our Q vectors with increasing energy restricts us to look for scattering between two disconnected surfaces of the α and β bands rather than for scattering within a single closed surface. By the same dispersion argument, interband scattering between concentric α and β sheets can also be excluded as a possibility since in general β disperses faster with increasing energy than α does (lengthening the Q with increasing energy).
a
c
Q3
Q2 Q1
b α β Q3 Q2 Q1 Figure C.1: a) Labeled Q vectors from QPI in comparison to Fermi surface suggested by theory b) and measured by ARPES c). Panel c) is reproduced from Ref. [86]. The QPI features seen likely originate from inside the yellow dashed box in either the Γ or Z planes. The key insight from QPI is the resolution of two features along (π, π) that may indicate additional folding of the β band Fermi surface or slight mismatch of sizes as the kz momentum is changed.
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In Table C.1,we convert the measured de Haas-van Alphen (dHvA) frequencies to Fermi surface areas in units of 1 Brillouin zone. The extremal areas of each band then give lower limits on the length of possible connecting Q vectors, which we estimate by assuming simple FS shapes consistent with theory. It is immediately apparent that the (π, π) Q vectors (Q2,3 = (0.24, 0.24), (0.27, 0.27)) can only come from the β band as the α band cylinders are too far separated in that direction. However, Q1 along (π, 0) may originate from either the α or β band as the measured Q1 = (0.29, 0) can originate on the zone edge where the two bands are close together. Table C.1: Estimated minimal Q vectors from dHvA measurements of CeCoIn5 . CeCoIn5 α β
C.2
dHvA Areas Up to 28% of BZ Up to 61% of BZ
dHvA Mass 8-18 40-50
Q’s along (π, π) >(.47,.47) rlu >(.22,.22) rlu
Q’s along (π, 0) >(.3,0) rlu >(.22,0) rlu
Phenomenological Modeling of Normal State Band Structure
To speculate on the qualitative features of QPI in the superconducting state, we first capture the energy dispersions of the normal state in an over-simplified 2-dimensional (2D) model. For concreteness, we identify two 2D surfaces (corresponding for example
110
to two different kz cuts of the band) whose energy dispersions k and χk are given by (kx , ky ) = µ + t1 (cos (kx ) + cos (ky )) + t2 cos (kx ) cos (ky ) + t3 (cos (2kx ) + cos(2ky )) χ(kx , ky ) = ν + s1 (cos (kx ) + cos (ky )) + s2 cos (kx ) cos (ky ) (C.1) + s3 (cos (2kx ) + cos (2ky )) (µ, t1 , t2 , t3 ) = (−31.3, 67.2, −124.6, 13.2) mV (ν, s1 , s2 , s3 ) = (−54.6, 36.3, 5.0, −9.2) mV. By suitable adjustment of the hopping parameters, the 2kf scatterings within this model can be made to reproduce the dispersions and general QPI pattern measured in the experiment as shown in Fig. C.2.
Figure C.2: (a,b) show the experimental QPI peaks dispersing along (π, 0) and (π, π) directions, respectively, overlaid with the dispersion of the appropriate 2kf scatterings calculated from the parameterized 2D (k) and χ(k) surfaces. The in-field H = 5.7 T data (non-superconducting) was substituted for the linecut at 0 energy. The temperature of the measurement was 245 mK. (c) shows the typical constant energy contour for (k) (blue solid) and χ(k) (blue dashed). The simulated Born scattering QPI pattern for 1.5 mV is shown in (d). 111
To calculate the QPI patterns, we applied the Born scattering approximation: 2 dI(q, ω) 2πe X ˆ ˆ S(q, ω) = = Nt tN (q, ω)tˆ i,j dV ~ i,j=1 Z 2 dk ˆ ˆ (q, ω) = − 1 Im ˆ + q, ω) N G(k, ω)Uˆ G(k π (2)2
(C.2) (C.3)
where S(q, ω) reflects experimental Fourier transform of the differential conductance. We take the density of states of the STM tip Nt = 1, tˆ = t0 t0χ with t = −1 and tχ = 0.7 denoting the propensity to tunneling into the and χ surfaces, and 0 ˆ = G 00 encoding the full Greens functions G0 (k, ω) = (ω + iΓ − (k))−1 and G 0 Gχ G0χ (k, ω) = (ω + iΓχ − χ(k))−1 . The lifetimes Γ and Γχ are taken to be 0.1 mV for U U the normal state, and the scattering matrix Uˆ = Uχ Uχ = .31 0.3 .7 . χ Finally, we note that the broad feature Q1 along (π, 0) seen in experiment may come from an overlap of intra-surface scattering from both the k and χk surfaces, and would be more precisely captured in a full 3D model that considers kz dispersion.
C.3
Superconductivity Gapping the Phenomenological Band Structure
We investigate how superconductivity qualitatively changes the QPI patterns by applying both a dx2 −y2 and dxy gap function on our model normal state band structure k and χk . In the presence of superconductivity, the Greens functions for k and χk acquire particle/hole channels given by the 2x2 matrices GS (k, ω) = ((ω + iΓ )I − (k)σ3 − ∆(k)σ1 )−1 and GSχ (k, ω) = ((ω + iΓχ )I − χ(k)σ3 − ∆(k)σ1 )−1 , where σi are
112
ˆ (q, ω) still hold with now the Pauli matrices. The above equations for S(q, ω) and N
(GS )11
(GS )11
0 0 0 (GSχ )11 0 (GSχ )12 ˆ= G (GS ) 0 0 (GS )22 21 0 (GSχ )22 , 0 (GSχ )21
−t 0 ˆ 0 −tχ , t = 0 0 0 0
0
0 0 0 t 0 0 tχ ,
(C.4)
and considering potential scattering
0 0 U Uχ Uχ Uχ 0 0 Uˆ = 0 0 −U −Uχ 0 0 −Uχ −Uχ
.
(C.5)
Using identical model parameters as in the normal state calculation (with the exception of Γ = Γχ = 0.05 mV), we simulate the experimental QPI at three energies for a dx2 −y2 gap on the k and χk surfaces (Fig. C.3 b) ,χ
∆ (kx , ky ) =
∆,χ x2 −y 2 2
(cos (kx ) − cos (ky ))
(C.6)
with ∆x2 −y2 = 0.67 mV and ∆χx2 −y2 = 2.55 mV such that the maximum gaps on and χ are 0.5 mV. The assumption of equal gaps on both and χ is an arbitrary feature of our model. Since our high resolution QPI data show subtle changes between the superconducting and normal states throughout Q-space (see Fig. 4.3 of main text), no unambiguous identification of completely ungapped portions of the FS can be made; likewise, determining the size of the gap on different surfaces based on these subtle features would be purely speculative. Alternatively, we can also consider a dxy gap
113
(Fig. C.3 c)
∆,χ (kx , ky ) = ∆,χ xy (sin (kx ) ∗ sin (ky ))
(C.7)
with ∆xy = 0.67 mV and ∆χxy = 1.09mV , maintaining a maximal 0.5 mV gap on both surfaces.
Figure C.3: Panel (a) shows the experimental data, while the bottom two panels show the resulting QPI pattern with application of a dx2 −y2 gap symmetry (b) and dxy gap symmetry (c) on our phenomenological band structure model. The measured data is more consistent with a dx2 −y2 gap. 114
Comparison of the panels shows that the experimental data cannot be reconciled with a dxy gap on the surface, and is qualitatively most consistent with a dx2 −y2 gap on both surfaces. However, such analysis cannot reproduce the strong electronhole asymmetry displayed by the data in Fig. 4.3 f)-o) of the main text, whose explanation may require additional assumptions about impurity effects or ungapped regions of the Fermi surface. These assumptions, absent independent experimental justification, together with the complex 3D, multi-band Fermi surface of CeCoIn5 , make extraction of the superconducting gap from QPI data ambiguous.
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Appendix D Details of Cd3As2 Landau Level Simulation In this appendix, we describe the four-band model Hamiltonian used in this thesis to capture the Landau level spectra of Cd3 As2 within the framework of prior ab initio calculations and ARPES measurements.
D.1
Modified Four-Band Kane Hamiltonian
An eight band Kane model is generally used to describe the low-energy band structure of small band gap semiconductors such as InSb and HgTe. Near the charge neutrality point, the description of the 3D Dirac semimetal phase can be further reduced to a Hamiltonian using the four-band basis of the S1/2 , 1/2 , P3/2 , 3/2 , S1/2 , −1/2 , P3/2 , −3/2 states. The total Hamiltonian is composed of two terms
H = Hef f + HZeeman ,
116
(D.1)
where Hef f models the band structure and HZeeman is the Zeeman energy. The effective Hamiltonian Hef f has the form
∗
Ak+ Dk− B (k) M (k) Ak− −M (k) B ∗ (k) 0 Hef f (k) = 0 (k) + Dk B(k) M (k) −Ak− + B(k) 0 −Ak+ −M (k)
(D.2)
where k± = kx ± iky , and the terms A, 0 (k), M (k) encode the electronic dispersion. We ignore the terms involving D, which describe any possible inversion symmetry breaking in the crystal structure, and B(k), which contains higher order terms allowed by crystal symmetry, since they are expected to produce only higher order corrections. The direction of the momentum kx , ky , and kz are associated with the a, b, and c axes of the crystal, respectively. However in the presence of a magnetic field, it is more convenient to describe the Hamiltonian in terms of the magnetic axis (k3 ) parallel to the field, because the momentums k1 and k2 perpendicular to the magnetic axis are quantized by the field. Accordingly, k1 and k2 can be transformed into the sum of ladder operators:
k1 = √
1 i (a† − a), k2 = √ (a† + a) 2lB 2lB
(D.3)
p (~/eB) is the magnetic length. The raising (lowering) operators a† (a) √ for the Landau levels |ni obey the usual relations: a† |ni = n + 1 |n + 1i, a |ni = √ n |n − 1i. We use a 3D rotation matrix U to transform the vectors (k1 , k2 , k3 ) of where lB =
117
the magnetic frame into the (kx , ky , kz ) of the crystal frame,1
kx k1 k1 k = U k = Rz (45◦ )Ry (φ = 54.7◦ )Rz (−45◦ ) k y 2 2 kz k3 k3
(D.4)
and numerically diagonalize the Hamiltonian in the |ni basis to compute the Landau level energy spectrum. In the above, we explicitly give the form of the matrix U to rotate [001] into [112] while keeping [100] and [010] symmetric about the [110] line. The semi-classical Landau quantization analysis shown in Fig. 5.3 c) demonstrates that electronic dispersion in the tilted (112) plane for the conduction band of Cd3 As2 is linear in an extended energy range from -100 mV to 300 mV. The original Hamiltonian proposed in Ref. [41] allows for both linear (via Ak± ) and quadratic terms in the dispersion in the kx or ky directions, but only a quadratic term in the kz direction. To reflect the linear dispersion in a wide energy range in all three momentum directions, we introduce a hyperbolic dispersion along the kz direction. 0 (k) = C0 + C1 kz2 + C2 (kx2 + ky2 ) q M (k) = M0 + M32 + M1 kz2 + M2 (kx2 + ky2 )
(D.5) (D.6)
We remark that this modified Hamiltonian is not the unique description of the data, but is the minimal modification that is consistent with the high energy linear dispersion and the quadratic band minimum at low energy. With the proper parameters, Hef f (k) results in inverted bands with two Dirac points located along the [001] direction, evenly separated from the Γ point and dispersing linearly in 3D momentum space away from the two Dirac points. 1
All rotation matrices in a series of compound rotation matrices still rotate with respect to the original frame of axes, not about the intermediate frame. The Rz and Ry matrices are the basic right hand rule rotation matrices.
118
The Zeeman term HZeeman has the form HZeeman (k) =
µB gs 0 (σ · B) ⊗ 2 0 gp
(D.7)
where µB is the Bohr magneton, σ are the Pauli matrices, and g(s(p)) is the effective g-factor for the S(P) band. The effect of gp on the electron-like Landau levels is negligible away from the band minimum where the S and P bands are well separated in energy. Hence, we can only reliably estimate gs from the data and take gp = 2. We note that the effective Hamiltonian Hef f (k) implies an induced orbital angular momentum which breaks the degeneracy of the bands in the presence of magnetic field. Landau level splitting caused by this orbital angular momentum strongly depends on the angle of magnetic field as shown in Fig. 5.5 b), while splitting caused by Zeeman term depends less on the angle. The numerical implementation of the above description is given in the source code below, the original version written by Itamar Kimchi and Andrew Potter of University of California, Berkeley. Here it has been streamlined to output a matrix of simulated energies (specifically at the Γ point, or k3 = 0) for a vector of magnetic fields B. Using below and by comparing with the experimental data matrix, we can minimize the error by least squares and determine a good set of parameters to reproduce the data. Generally, the inability of the experimental data to constrain details near the Dirac point due broadening leads to local minima in the least squares fit, stemming from the flexibility of parameters related to the low energy features. However, the well-defined vF given by the semiclassical analysis must be still reproduced in this model calculation, so the combined effect of the kx and kz dispersions at high energy is well constrained. Moreover, splitting of the levels in a B-field naturally arises. In short, the model illustrates the features of the data are consistent with the original theoretical model of inverted bands. 119
Listing D.1: Appendix2/LL Cd3As2.m: Landau Level Simulation Code % % Computes LL ’ s of Cd3As2 and DOS , using linearized low - energy Hamiltonian function [ out all_levels ] = LL_Cd3As2 ( param , x , levels , cutoff , q ) % This function can be used with lsqcurvefit to produce a matrix of % simulated levels " out " , where the columns are the B fields and the rows % are the levels . % param - parameters for the hyperbolic Hamiltonian % x - one column vector of the required B fields % levels - an integer specifying how many levels you want in the output % cutoff - start counting levels above this minimum energy % q - regularization coefficient , needed to move spurious levels away from % energy range of interest N_max = 36; % number of oscillator levels to keep ( truncate Hilbert space at High LL index ) . N_vals = 4*( N_max ) ; % if full diagonalization % 3 - dispersion ( dispersion along B field direction ) % Here we are only interested in the minimum of the Landau band at k3 = 0. N_k3 = 1; k3s = 0; dk3 = 0; % % band structure parameters M0 = param (1) ; M1 = param (2) ; M2 = param (3) ; inversion = param (4) ; M3 = -M0 - inversion ; E_dirac = param (5) ; C1 = param (6) ; C2 = param (7) ; A = param (8) ; g_e = param (9) ; phi_degrees = param (11) ; g_h = param (10) ; % %%%%%%%%%% Constants for now %%%%%%%%%%%% Delta = 0; % Bulk inversion asymmetry % %%% Compute C0 to keep Dirac Point at 0 energy . kd is set by above . kd = real ( sqrt (( M0 ^2 - M3 ^2) / - M1 ) ) ; C0 = C1 * kd ^2; % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
120
% % Rotating crystal coordinates x ,y , z to magnetic field coordinates 1 ,2 ,3 % ( magnetic field is along \ hat {3}) phi = pi /180* phi_degrees ; % the following rotation matrix rotates % essentially Ry (45) Rz ( phi ) Ry ( -45) U = ... [1/2*(1+ cos ( phi ) ) 1/2*( -1+ cos ( phi ) ) 1/2*( -1+ cos ( phi ) ) 1/2*(1+ cos ( phi ) ) -( sin ( phi ) / sqrt (2) ) -( sin ( phi ) / sqrt (2) )
(0 ,0 ,1) towards (1 ,1 ,2) :
sin ( phi ) / sqrt (2) ;... sin ( phi ) / sqrt (2) ;... cos ( phi ) ];
% % LL Raising and Lowering Operators adag = sparse ((2: N_max ) ,[1:( N_max -1) ] , sqrt (1: N_max -1) , N_max , N_max ) ; a = sparse ([1:( N_max -1) ] ,(2: N_max ) , sqrt (1: N_max -1) , N_max , N_max ) ; n_hat = adag * a ; % truncate landau level tower by coupling weyl nodes above a large n: id_k = speye ( N_max ) ; large_n_reg = n_hat -(( N_max -2) .* id_k ) ; % % Construct Hamiltonian % generic pauli matrices : pauli_z = sparse ([1 0; 0 -1]) ; pauli_x = sparse ([0 1; 1 0]) ; pauli_y = sparse ([0 -1 i ; 1 i 0]) ; pauli_0 = speye (2) ; % Pauli Matrices ( orbital ) mu_z = pauli_z ; mu_x = pauli_x ; mu_y = pauli_y ; mu_0 = speye (2) ; mu_up = ( mu_0 + mu_z ) ./2; mu_down = ( mu_0 - mu_z ) ./2; % orbital g - factor % Note that g_h is defined as a multiplier on g_e inorder to keep them % the same sign when using lsqcurvefit . mu_g = g_e .* mu_up + ( g_h * g_e ) .* mu_down ; % Pauli Matrices ( spin ) s_z = pauli_z ; s_x = pauli_x ; s_y = pauli_y ; s_0 = speye (2) ; % % Construct Hamiltonian and Compute Spectrum for each k3 eigenvalue Evals = zeros ( N_k3 , N_vals ) ; H_reg = q * sqrt (2* N_max ) * kron ( kron ( s_x , mu_x ) , heaviside ( large_n_reg -0.5) ) ;
121
% H_reg should regularize the theory by coupling the two weyl nodes at % large landau level .
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculate all terms outside of the B field loop and scale by B field only % inside the loop for speed . Below are the " zero - field " terms . k2_0 = 1/ sqrt (2) .*( adag + a ) ; % k2 - momentum operator ( guiding center shifted to x =0) k1_0 = 1 i / sqrt (2) .*( adag - a ) ; % k1 - momentum operator % Rotate axes from along field to along crystal axes kx_0 = U (1 ,1) * k1_0 + U (1 ,2) * k2_0 ; ky_0 = U (2 ,1) * k1_0 + U (2 ,2) * k2_0 ; kz_0 = U (3 ,1) * k1_0 + U (3 ,2) * k2_0 ; H_Z_0 = 1/2*(5.8 e -5) *... kron ( kron ( U (1 ,3) .* s_x + U (2 ,3) .* s_y + U (3 ,3) .* s_z , mu_g ) , id_k ) ; H_A_0 = A .* kron ( kron ( s_z , mu_x ) , kx_0 ) A .* kron ( kron ( s_0 , mu_y ) , ky_0 ) ; out = zeros ( size ( levels ,1) , size (x ,1) ) ; all_levels = zeros ( N_vals , size (x ,1) ) ; % Loop over B - field . for i_B = 1: size (x ,1) B = x ( i_B ,1) ; % LL Lowering and raising op ’ s magnetic_length = (256.6/ sqrt ( B ) ) ; % 257 A / sqrt ( T ) converted to unit - cell size % Rotate axes from along field to along crystal axes kx = (1/ magnetic_length ) * kx_0 ; ky = (1/ magnetic_length ) * ky_0 ; kz = (1/ magnetic_length ) * kz_0 ; % construct the Hamiltonian E_k = kron ( kron ( s_0 , mu_0 ) , C0 * id_k - C1 * kz ^2 - C2 *( kx ^2+ ky ^2) ) ; H_z = kron ( kron ( s_0 , mu_z ) , M0 * id_k + sqrtm ( full ( M3 ^2* id_k M1 * kz ^2) ) - M2 *( kx ^2+ ky ^2) ) ; H_A = (1/ magnetic_length ) * H_A_0 ; H_Z = B * H_Z_0 ; H_BIA = Delta .*( kron ( kron ( s_y , mu_y ) , id_k ) ) ; H = ( H_z + H_A + E_k + H_Z + H_BIA + H_reg ) ; Evals = sort ( real ( eig ( full ( H ) ) ) ) + E_dirac ; all_levels (: , i_B ) = Evals ’;
122
% We want to compare with only the levels that we have data for . In % other words the first ( N = levels ) levels above some cutoff energy . ind = find ( Evals > cutoff ,1 , ’ first ’) ; levels_out = ( ind -1) + levels ; out (: , i_B ) = Evals ( levels_out ,1) ’; end
To find the proper parameters for our model, we first fix the value of A, M2 , and C2 (related to kx -ky dispersion) to agree with the photoemission data measured in the (001) plane of Cd3 As2 samples [108] from the same sample grower. The remaining parameters (summarized in Table 1) are chosen to reproduce the observed Landau level peaks. These parameters are used for the plots in Fig. 5.5 a, b, d, and e. Due to the absence of the valence band Landau levels, we cannot obtain a precise determination of the size of the band inversion (20 mV is used in our model). However, the general behavior of the Landau level structure, such as the diminishing two-fold splitting and linear high energy dispersion, is independent of this quantitative detail. Figure D.1 a) and b) show the band dispersion for the two axes along kx and kz passing the Dirac points. The band dispersion measured by angle resolved photoemission is well reproduced in this model with the above parameters as shown. The band inversion and two 3D Dirac points are revealed. Table D.1: Parameters for the modified four-band model for Cd3 As2 . C0 (eV) = -0.219 C1 (eV ˚ A2 ) = -30 C2 (eV ˚ A2 ) A (eV ˚ A) = 2.75 gs = 18.6
M0 (eV) = -0.060 M1 (eV2 ˚ A2 ) = 96 M2 (eV ˚ A2 ) = 18 M3 (eV) = 0.050 gh = 2
123
b 0.2
0.2
0
0
Energy (eV)
Energy (eV)
a
-0.2
-0.4
-0.4 -0.2
-0.2
0
-0.05
0.2
(1/A)
0
0.05
(1/A)
Figure D.1: a), b), The band dispersion along kx axis (a) and kz axis (b) passing the Dirac point for parameters used to simulate the Landau level peaks. The Fermi velocity along the kx direction is 5.1 eV ˚ A(= 7.6 · 105 m/s), in agreement with prior ARPES measurements.
D.2
Schematic Demonstration of the Weyl Fermion and the Low Field Regime
As discussed in the main text, the Weyl fermion can be realized in Cd3 As2 when the magnetic field is applied along the [001] direction. To schematically demonstrate the Weyl fermion, we simulate the Landau levels using the Hamiltonian and parameters provided in the prior calculation paper. The degenerate Weyl fermions with two different chiralities are separated in momentum space when the magnetic field is applied along the [001] direction as shown Fig. D.2 a-b). The crossing points are shifted away from the original Dirac points as field strength is increased (Fig. D.2 b). Landau levels of non-zero index (n > 1) are formed below the Lifshitz energy for a relatively small magnetic field (Fig. D.2 a). For these levels, extrema in the energy dispersion of the Landau level band lead to two singularities in the DOS: one from maximum at the gamma point and another from minima near the Dirac points. However, the high index Landau levels formed above the Lifshitz transition have only 124
= 54.7
(1/A)
0
−10 −0.05
0
(1/A)
0
−10 −0.05
0.05
d
0.05
Weyl points
E Dirac
Weyl points
20
E Lifshitz
=0
B= 12T ,
= 54.7
0
0.05
0
0.05
(1/A)
E Lifshitz
B= 1T ,
0
B= 12T ,
E Dirac
20
Energy (meV)
E Lifshitz E Dirac Weyl points nearly overlapping at Dirac points
E Lifshitz
Energy (meV)
20
0
−10 −0.05
c
b
=0
E Dirac
Energy (meV)
20
B= 1T ,
Energy (meV)
a
0
−10 −0.05
(1/A)
Figure D.2: a), b), Simulated Landau levels with the magnetic field applied along the [001] direction for B = 1 T (a) and B = 12 T (b). The Weyl points and the energy of the Lifshitz transition are marked. c), d), Simulated Landau levels with the magnetic field applied along the [112] direction for B = 1 T (c) and B = 12 T (d). The lowest Landau levels have a gap at the Dirac points due the C4 symmetry breaking. The momentum k is along the magnetic field direction. one DOS singularity from the gamma point. In the measured spectra, all the peaks originate from the singularity at the gamma point due to the shallow band inversion as modeled in Fig. 5.5 e). For the purpose of comparison, calculated Landau levels with non-zero field angle are plotted in D.2 c-d). Moreover, the lowest Landau levels open a gap at the crossing points due to the broken C4 symmetry.
125
Appendix E Charge Ordering in Underdoped Bi2Sr2CaCu2O8+δ in a Magnetic Field An overwhelming amount of experimental evidence, dating from the discovery of stripe-like spin and charge order in coincidence with the suppression of Tc in the La2−x Srx CuO4 and La2−x Bax CuO4 family [128], has now established that superconductivity in the high-Tc copper-oxides competes with an alternative electronic order, such as spin or charge density wave. Most recently, charge order was discovered in the YBa2 Cu3 O6+x (YBCO) compounds near 1/8 doping using resonant x-ray techniques [129]. In Ref. [130], the intensity of this order was shown to be strongly magnetic field dependent below Tc , with an enhancement by a factor of 3 in a modest magnetic field of 15 T. Subsequently, Refs.[131] and [132] extended the ubiquity of this ordering phenomena to the Bi-based cuprate family, in bilayer Bi2 Sr2 CaCu2 O8+δ (Bi2212) and single layer Bi2 Sr2 CuO6+δ (Bi2201), respectively. Therefore, a natural question to raise is whether the magnetic field enhancement of charge order in the YBCO family similarly affects charge order in the Bi-based family. Combined with the tantalizing 126
evidence that charge modulations of wavelength 4 a0 are enhanced inside of the vortex cores of slightly overdoped (Tc = 89 K) Bi2212 at 7 T [133], the presumptive answer might be a resounding ‘yes’. If so, as the field is increased, does the charge order spread in space through the extent and number of vortex cores, or does the intensity of the existing charge order at zero-field simply become enhanced? In the summer of 2013, the author, with the help of Mallika Randeria, attempted to answer these questions in DRSTM, applying a maximal field of 14 T.
E.1
Experimental Results
Two underdoped samples of Bi2 Sr2 CaCu2 O8+δ were used in this study: GG0904 with Tc = 76 K and GG0959 with Tc = 58 K. Inverting the empirical relation between Tc and hole doping δ,
Tc = Tc,max · (1 − 82.6(δ − 0.16)2 )
(E.1)
with Tc,max of Genda Gu’s optimally-doped samples estimated to be 90 K, we obtain δ = 0.12 for the Tc = 76 K sample, and δ = 0.094 for the Tc = 58 K sample. Hence, we are studying samples near 1/8 doping and thus can directly compare with the YBCO studies.1 The highest quality data was obtained for the Tc = 58 K sample, but the qualitative results are similar for both samples. In Fig. E.1, we show an area on the UD 58 K sample over 20 b-axis supermodulations wide, and of length 75% of the width. This area of the sample was tracked from 1 T to 14 T with the same tip at a temperature of 2.5 K.2 The spatially averaged spectra show the robustness of the high-Tc materials to a magnetic field. Indeed, 1
The magnetic field enhancement effect is not expected to be restricted to 1/8 doping, but just perhaps strongest there. 2 We took data at 1 T to take advantage of slightly better noise performance due to magnetic damping; however 1 T is not expected to be different than 0 T.
127
UD 58 K
PG
e-b coupling?
530 x 400 Å
SC
Figure E.1: The same area of a Bi2212 (UD 58 K) sample tracked from 1 T to 14 T. The spatially averaged spectra show suppression of the pseudogap coherence peaks (labeled in cyan) at high field. Moreover, the depth of the SC gap (labeled in purple) is filled in at high fields. Finally, more speculative features (orange) outside of the pseudogap, reminiscent of the electron-boson coupling ‘wiggle’, is reduced in amplitude at high field. since the upper critical fields for Bi2212 are in excess of 150 T, the modest 14 T field applied by our superconducting magnet barely changes the spectra. Division of the 1 T spectrum by the 14 T spectrum reveals more clearly the resolvable changes. First, lets us address the change in the larger, dominant gap in spectrum with energy scale ±50 mV, known as the ‘pseudogap’ in underdoped cuprates. We see that its ‘coherence’ peaks are broadened by the field. Second, the zero bias conductance and the conductance in vicinity near zero bias (the Fermi level), set by the superconducting gap, are enhanced by up to ∼15% at 14 T, showing a increase in the number of nodal quasiparticles in the d-wave superconductor due to vortex/field line penetration. Finally, we notice a wiggle in ratio of the spectra at large positive biases outside the pseudogap. Traditionally, wiggles in the spectra outside the gap for optimally and
128
overdoped Bi2212 samples have been attributed to electron-boson coupling, where the boson is of electronic, rather than lattice origin [134]. Such features have not been extensively studied on the underdoped side; however, it does appear that the magnetic field also damps such features and may tune the underlying electron-boson coupling. One criticism of these interpretations is that ‘magnetic’ broadening due to Zeeman-type physics naturally broadens any feature in the spectra. However, an estimation with g = 2 gives an energy scale of ∼1.4 mV at 14 T, and it appears that the experimental changes in the spectra with field occur over much larger ∼10 mV scales; hence, these changes may not be directly attributable to simple broadening. Of course, this crude argument needs to be further vetted in quantitative simulation.
E.2
Energy and Spatially-Resolved Density of States in a Magnetic Field
The next natural question to pose is to what extent the energy-resolved spatial modulations change in a field, either through the introduction of identifiable vortex cores or the widespread enhancement of charge order. Figure E.2 shows several the real space conductance maps taken from -49.5 mV to +49.5 mV in 5.5 mV increments. One immediately notices the almost identical features in the 1 T and 14 T maps; that if the maps were mixed up, one would be hard pressed to sort out which one was taken at 1 T and which one taken at 14 T. Indeed, it appears that the dramatic 300% magnetic field enhancement of charge order seen in YBCO at 15 T is not so obviously present in Bi2212 at 14 T, at least on the surface, visualized by STM measurement. From first glance, any changes resolved by more quantitative analysis would likely be restricted in magnitude to the order of the change in the spectra, shown in Fig. E.1, that is to say around ∼10 %. Moreover, as the conductance maps at 1 T and
129
14 T match almost feature to feature,3 it does not appear that charge order spreads in the sample, but that the existing charge ordering at 1 T is amplified in intensity (especially for high biases). B = 14T
B = 1T
B = 14T
E = 33 mV
E = -11 mV
E = 22 mV
E = -22 mV
E = 11 mV
E = -33 mV
E =0 mV
E = -44 mV
B = 1T
Figure E.2: Real space resolved spectroscopic mapping of the electronic states in Bi2212 UD 58K at 1 T and 14 T. The same color scale is used for both fields. The nearly identical features at all energies for both fields show that charge order does not in general spread in the sample due to the increased magnetic field. Rather, there does appear to exist small patches (e.g., circled in white for E = +22 mV) where the existing charge ordering appears stronger in a magnetic field; however, it is difficult to definitively rule out very minute differences in tip and noise condition that may affect the outcome of the measurement. At low biases, (e.g. E = 0 mV), there does appear to be some areas where charge modulations appear from previously absent areas. In any case, Fourier transforms of these maps can give a more quantitative estimate of the enhancement. 3
Upon revisiting this data more carefully during the writing of this appendix, it appears that the correlation between the maps at 1 T and 14 T is weakest at low energy scales, namely between ±5.5 mV.
130
Comparison of the Fourier transforms of the energy-resolved spectroscopic maps provides a more quantitative estimate of the enhancement of charge ordering in a magnetic field. In Fig. E.3, we show the cut along the Bragg direction (π, 0), denoted by the red arrow in the FFT of E = 22 mV, as a function of energy. It is apparent that the charge ordering is a little stronger in a magnetic field, as the subtraction of the two panels shows enhancement near E = 22 mV, circled in magenta. The linecut at E = 22 mV shows the intensity of the charge ordering wave vector Q∗ = (0.25, 0) · 2π/a0 to be enhanced by approximately 11%, which seems reasonable as this is on the order of magnitude of the changes that we see in the spectra. Thus, the magnetic field likely does not precipitate an extraordinary or disproportional response in Bi2212, but produces a more benign response proportional to the weakening of superconductivity (recall that the zero bias conductance was also elevated by ∼15%). This contrasts with the case of YBCO, where a 300% enhancement of the charge order was seen for fields that hardly affect superconductivity in the sample. A further complication to the interpretation of the results presented here is the presence of the QPI wave vector Q1 , which is nearly coincident with Q∗ . Hanaguri’s seminal measurements on another cuprate family superconductor, the oxychloride Ca2−x Nax CuO2 Cl2 , showed the vector Q1 to be enhanced in a magnetic field due to coherence factor effects for magnetic scattering [135]. To elaborate, QPI wavevectors that connect parts of the Fermi surface with the same sign of the gap function (Q1 , Q4 , Q5 ) are enhanced by scattering off magnetic vortices, while QPI wavectors that connect parts of the Fermi surface with opposite sign of the gap function (Q2 , Q3 , Q6 , Q7 ) are alternatively weakened. Thus, to prove that charge order is enhanced, one must separate the effect on Q∗ from that on Q1 , which is also expected to be enhanced by a magnetic field. Ideally, one must show that degree of enhancement in the vicinity of Q1 /Q∗ is significantly stronger than the enhancement of Q4 , Q5 ; however, the data does not clearly show the other Qi for such underdoped samples. In 131
1T a)
1T
14 T
c)
14 T
14 T - 1 T d)
f)
Hanaguri’s Result on Ca2-xNaxCuO2Cl2
Q* or Q1?
14 T - 1 T b)
e)
Peak Height 364 328
Figure E.3: a) The wavevector intensity cuts along (π, 0) as a function of energy for 1 T and 14 T. b) Difference of the wavevectors cuts showing an enhancement of the Q∗ charge ordering feature centered around 20 mV. c) Typical FFTs of the spatial patterns shown in Fig. E.2 at 1 T and 14 T. d) Difference of the FFTs showing enhancement in the region near Q∗ /Q1 . Unfortunately the other octet model Qi ’s are not well resolved for such underdoped samples. e) The single energy cut along (π, 0) at E = 22 mV showing an approximately 11% enhancement at 14 T over 1 T. f) Hanaguri’s data [135] on an oxychloride superconductor showing a phase-sensitive enhancement of the octet model Qi ’s. Note that the map shown in his data is a so called Z-map. His paper makes no mention of whether the strong checkerboard charge order in that material is enhanced in a magnetic field. this case, one can only comment that the enhancement of Q1 /Q∗ region appears to be particle-hole asymmetric, favoring the particle side, which suggests that it is indeed Q∗ that is enhanced. Enhancement of Q1 is in principle particle-hole symmetric, while the charge order Q∗ is centered at positive energies, around positive 20 mV. 132
A final thought on the difference between the results obtained in YBCO and in Bi2212 is whether it could be attributed to bulk vs. surface probes. Could charge order be enhanced in the bulk, but not on the surface? Only time will tell whether this magnetic field phenomena is truly ‘ubiquitous’ across the cuprate landscape, since these x-ray experiments in a magnetic field should be readily doable on the Bi2212 samples.
E.3
Where are the Vortices?
The initial apparent observation was made that no static vortices could be imaged in underdoped Bi2212. In the process of looking at the data more carefully during the writing of this thesis, there probably exist static vortices, with at least one definitive identification, although they are very difficult to identify. In overdoped and optimally doped Bi2212 samples, vortices pinned by strong disorder (sometimes from intentional Zn dopants) could be imaged clearly from the conductance map at Ekink ∼ ±7 mV [62]. Whether the vortices are static is an important experimental question, necessary to reconcile with the fact that resistivity measurements show these samples to have zero resistance at the temperatures and fields of the STM measurement, suggesting pinned vortices. At a given field B, we expect B/Φ0 vortices per unit area, where Φ0 = 2.1 · 10−15 T m2 . In other words, at 14 T, one expects a single vortex to occupy 15000 ˚ A2 (or a square of side length 122 ˚ A). Hence in our field of view of 530 ˚ A by 400 ˚ A, we expect approximately 14 vortices at 14 T, and accordingly approximately one vortex at 1 T. As mentioned before the STM conductance maps show the strongest change between 1 T and 14 T for energies near the Fermi energy, i.e. E = -5.5, 0, and 5.5 mV (near Ekink ). Summing these conductance maps to gain the strongest signal, we can compare the two fields in Fig. E.4 to identify several regions of enhanced 133
conductance, also containing new charge modulations. We tentatively identify these regions as vortices. However, the number is insufficient to account for all of the approximately 13 extra vortices that are expected to thread the field of view as the field is dialed up from 1 T to 14 T. More careful analysis is currently being investigated to drift correct the maps and enable pixel-to-pixel real space subtraction; however, if the simple ‘eye’ test cannot pickout the vortices, they likely they are of fundamentally different character than the vortices in near optimally doped Bi2212. Qualitatively, our result is similar to Ref. [133] in that 4 a0 charge ordering accompanies the vortex regions. However, the analysis by Fourier transform nevertheless does not reveal a strong quantitative enhancement. Perhaps this suggests that the metric for STM is somehow not directly comparable to the metric of x-ray scattering; or that the background charge modulation for underdoped Bi2212 is already too dense at zero field, leaving little room for charge order to spread on the surface.
134
1 T ~ 1 vortex
14 T ~ 14 vortices
Figure E.4: These maps are the sum of the conductance maps at the three lowest energies: -5.5 mV, 0 mV, 5.5 mV. The higher energy maps show much less variation with field. In this sized area of the sample, an additional 13 vortices should penetrate at the higher 14 T field. We can identify at least 3 regions where the spatial electronic patterns change significantly. The lowest one near the center of the field of view is strongly suggestive of an vortex core, since an area with previously low conductance now possesses high conductance in field. Moreover, charge modulations appear inside this presumptive vortex core. However, the number and intensity of the changes over the entire field of view seems to be a bit lacking. Therefore, if indeed additional vortices exist in this field of view, it appears they are well ‘camouflaged’ in the background charge order and require exquisitely low noise measurement and drift correction to see.
135
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